Definition of Link:

A link is basically one part of a machine that connects with other parts. These parts together create movement. For example, in a bike chain, each segment of the chain can be considered as a link.

Key Points:

• Rigid Body: A solid part that doesn’t change shape.
• Connects to other links: Through joints like pins or screws.
• Allows Motion: The connected parts can move in specific ways relative to each other.

Classification of Kinematic Links

1. Rigid Link:

• Definition: A rigid link is a solid part that doesn’t change shape or bend when force is applied.
• Applications:
  • Bicycle Frame: The frame stays strong and transfers your effort when you pedal.
  • Car Engine (Crankshaft): The crankshaft converts the piston’s movement into rotational motion to move the car.
  • Scissors: The blades stay firm and help cut things cleanly without bending.

2. Flexible Link:

• Definition: A flexible link can bend or flex, but still transfers motion or force.
• Applications:
  • Belt Drive in a Machine: The belt connects pulleys and bends as it moves, transferring motion between them.
  • Crane Cables: The cables bend to support and lift heavy loads in construction without breaking.
  • Elevators: The steel cables flex while pulling the elevator up and down safely.

3. Fluid Link:

• Definition: A fluid link uses liquids or gases to transfer force or motion.
• Applications:
  • Hydraulic Brakes in Cars: Pressing the brake pedal moves fluid, which presses the brake pads against the wheels to stop the car.
  • Hydraulic Jacks: Lifts heavy objects like cars by using fluid pressure.
  • Pneumatic Tools: Air-powered tools like drills or hammers use compressed air to create motion.

Types of Relative Motion:

Completely Constrained Motion: The object is fully restricted to move in just one direction (e.g., a piston in a cylinder).

Incompletely Constrained Motion: The object can move in more than one direction (e.g., a shaft in a hole).

Successfully Constrained Motion: External forces or design make sure the object moves in the desired direction (e.g., a foot pedal).

1. Constrained Motion:

• What it is: This type of motion occurs when the movement of a body is restricted or limited in some way.
• How it works: The object can only move in a specific path or direction because of physical restrictions like joints, tracks, or links.
• Example:
  • Sliding Door: It can only move back and forth along its rails (restricted motion).
  • Piston in an Engine: It can only move up and down inside the cylinder (linear motion).

2. Unconstrained Motion:

• What it is: This is when the motion of an object is not restricted and it can move freely in any direction.
• How it works: The object can move without limitations in terms of path or direction.
• Example:
  • A Ball in Open Space: It can move in any direction when thrown.
  • A Car on an Open Ground: It can move in any direction, forward, backward, or sideways, as long as there’s nothing stopping it.

Classification of Kinematic Pair

1. According to Type of Relative Motion
2. Revolute Pair (Turning Pair)

• Definition: Allows one link to rotate around a fixed axis relative to the other link.
• Example: Door hinge where the door rotates around the hinge.

1. Prismatic Pair (Sliding Pair)

• Definition: Allows one link to slide or move linearly along a fixed path relative to the other link.
• Example: Piston in a cylinder which slides up and down.

1. Screw Pair

• Definition: Allows motion that combines both rotation and translation. One link rotates while moving linearly along its axis.
• Example: Lead screw in a lathe, which rotates and moves along its axis.

1. Spherical Pair (Rolling Pair)

• Definition: Allows one link to rotate freely in three-dimensional space around a fixed point.
• Example: Ball-and-socket joint in the human shoulder.

2. According to Type of Contact

Lower pairs and higher pairs are types of kinematic pairs that describe how two elements are joined together:

1. Lower Pair

• Definition: The contact between links is surface-to-surface, where the motion is constrained by surfaces in contact.
• Example:
  • Revolute Pair: Door hinge, where the surfaces of the hinge and door make contact.
  • Prismatic Pair: Piston in a cylinder, where the piston’s outer surface slides inside the cylinder.

1. Higher Pair

• Definition: The contact between links is point-to-point or line-to-line, which means the contact is not surface-based but occurs at discrete points or along lines.
• Example:
  • Screw Pair: Lead screw, where the contact occurs between the threads of the screw and the nut.
  • Gear Pair: Gears in a gearbox, where the contact happens between the gear teeth.

1. Wrapping Pair

• Definition: A wrapping pair involves a link that wraps around another link, where the contact is along a curved surface rather than a flat or point contact. This type of pair allows the wrapping link to move while maintaining contact around the other link.
• Example:
  • Belt and Pulley: In a belt drive system, the belt wraps around the pulley, transferring motion from one pulley to another.
  • Chain and Sprocket: A chain wraps around the sprocket wheels, allowing rotational motion to be transmitted.

3) According to Type of Closer:

1. Closed Pair/ Permanently closed pair:

When links are connected permanently such that during motion they will never get separated due to connection itself such a pair called as closed pair.

1. Unclosed / Forcefully Closed Pair:

When there is no mechanical connection between link. They are just kept together with the help of some external force during the motion, such a some external force during the motion, such a pair is called unclosed/ forcefully closed pair.

Spring—always in compression.

Spring is not a link here.

Type of Joints

1. Binary Joint:

• Definition: A binary joint is a joint where two links are connected.
• Example: A simple hinge or pin joint connecting two links, like a piston connected to a crankshaft.

2. Ternary Joint:

• Definition: A ternary joint is a joint where three links are connected at a single point.
• Example: In a slider-crank mechanism, the point where the connecting rod attaches to both the crank and the slider forms a ternary joint.

3. Quarternary Joint:

• Definition: A quaternary joint is a joint where four links are connected at a single point.
• Example: In more complex mechanisms like a four-bar linkage, where four different links meet at a common joint, you have a quaternary joint.

Types of Link:

Links are the rigid components that connect and move relative to each other. Links are classified based on the number of joints they are connected to. Here are the types of links:

1. Binary Link:

• Definition: A binary link is a link that is connected to two other links through joints.
• Example: A simple connecting rod in a slider-crank mechanism, where one end is connected to the piston and the other end to the crankshaft.

2. Ternary Link:

• Definition: A ternary link is a link that is connected to three other links through joints.
• Example: In a four-bar linkage, the input or output link might connect to three other links at different joints, forming a ternary link.

3. Quaternary Link:

• Definition: A quaternary link is a link that is connected to four other links through joints.
• Example: In complex mechanisms like some mechanical linkages, a quaternary link can connect to four other links to form the central part of the system.

Grubler’s Mobility Criteria

Grubler’s mobility criterion is used to determine the degree of freedom (mobility) of a mechanism, which is the number of independent movements it can perform. The formula is particularly useful for planar mechanisms. Here’s a breakdown of the criteria in points:

1. Formula:

M=3(L−1)−2J

• L: Number of links, including the fixed link.
• M: Mobility or degrees of freedom of the mechanism.
• J: Number of lower pairs (such as revolute or prismatic joints).

2. Interpretation:

• If M = 0, the mechanism is statically determinate (it doesn’t move).
• If M > 0, the mechanism has that many degrees of freedom.
• If M < 0, the system is over-constrained (too many constraints, not functional).

3. For Planar Mechanisms:

• Each link in a planar mechanism has 3 degrees of freedom (two translations and one rotation).
• Joints (lower pairs) restrict degrees of freedom. Each revolute or prismatic joint removes 2 degrees of freedom (since it only allows 1 type of motion).

4. Application:

• Grubler’s criterion is widely used to check the mobility of common planar mechanisms, such as the four-bar linkage.

Simple Mechanism:

1. Four-Bar Mechanism

The four-bar mechanism is one of the most basic and commonly used mechanisms in machines. It consists of four rigid links (bars or rods) connected by four revolute joints (hinges or pivots) that allow rotation.

Components:

• Crank: The input link that usually rotates.
• Coupler: The middle link that connects the crank to the follower.
• Follower: The output link that moves in response to the crank.
• Fixed Link: The link that remains stationary (doesn’t move).

Example:

• Wiper of a Car: The mechanism that moves the wiper back and forth is often a four-bar mechanism. When the crank rotates, the wiper moves in an arc, thanks to the follower.

Motion:

• The crank rotates, which causes the follower to move in either circular or back-and-forth motion.

2. Single Slider-Crank Mechanism

The single slider-crank mechanism is a more specific form of a four-bar mechanism. It is used to convert rotary motion into linear motion or vice versa.

Components:

• Crank: The rotating part (like in an engine).
• Connecting Rod: A link that connects the crank to the slider.
• Slider: A part that moves back and forth (slides) in a straight line.
• Fixed Link: The part that holds everything together.

Example:

• Piston in an Engine: In a car’s engine, the piston (slider) moves up and down in a straight line, and this motion is converted into the rotational motion of the crankshaft via the connecting rod.

Motion:

• The crank rotates, causing the connecting rod to move, which in turn makes the slider move back and forth in a linear path.

3. Double Slider-Crank Mechanism

The double slider-crank mechanism has two sliding pairs (sliders). It is mainly used when we need to convert rotary motion into linear motion in two directions.

Components:

• Two Sliders: These two sliders move back and forth in straight lines.
• Link: A rod or connecting bar that connects these sliders.
• Crank: A rotating part that provides motion to the mechanism.

Example:

• Elliptical Trammel: This is a device used to draw ellipses. It has two sliders that slide along two perpendicular axes (X and Y axes). When you rotate the link between them, it draws an elliptical path.

Motion:

• The rotation of the link between the two sliders causes both sliders to move in straight lines along their respective paths (usually along the X and Y axes).

Inversion:

In kinematics of machines, inversion refers to the process of changing the frame of reference in a mechanism by fixing a different link in the system while keeping the overall structure the same.

In simple terms, you have a mechanism with several links connected to each other (like in the four-bar or slider-crank mechanisms), and by fixing different links, you get different types of motion and mechanisms. Each new configuration, where a different link is fixed, is called an inversion.

In a four-bar mechanism, you have:

1. Crank: A link that can rotate fully (360 degrees).
2. Rocker: A link that can only oscillate or swing back and forth (less than 360 degrees).
3. Coupler: The link that connects the crank to the rocker.
4. Fixed Link: The part that remains stationary.

When you change which link is fixed, you get different types of motion, and that’s where these terms come in.

1. Crank-Crank Mechanism (Double-Crank Mechanism):

In this configuration, both the input link and the output link can rotate completely (360 degrees). This means you have two cranks.

• Crank-Crank Configuration happens when the shortest link (called the “crank”) is fixed. This allows the other two adjacent links to rotate completely.

Example:

• Bicycle Chain Mechanism: The pedals (crank) rotate fully, and the back sprocket also rotates fully, so it’s a crank-crank mechanism.

Motion:

• Both links rotate continuously in full circles.

2. Crank-Rocker Mechanism:

In this configuration, the input link is a crank (it can rotate fully), but the output link is a rocker (it oscillates back and forth). This is one of the most common inversions of the four-bar mechanism.

• Crank-Rocker Configuration happens when the shortest link is adjacent to the fixed link. The crank rotates fully, but the rocker swings back and forth.

Example:

• Windshield Wiper Mechanism: The motor rotates the crank, but the wiper blade only moves back and forth (it doesn’t rotate fully).

Motion:

• The crank rotates 360 degrees, and the rocker swings back and forth within a limited range.

3. Rocker-Rocker Mechanism (Double Rocker Mechanism):

In this case, both the input and output links are rockers, meaning neither of them can rotate fully. They both just oscillate back and forth.

• Rocker-Rocker Configuration happens when the longest link is fixed. The other two links will act like rockers and can only swing in limited angles.

Example:

• Folding Chairs: The two arms of the chair swing or rotate within a limited range, just like rockers.

Motion:

• Both rockers move back and forth (oscillate), but neither completes a full rotation.

Mechanism of Wiper

 Grashof’s Law

Grashof’s Law States:

In a four-bar mechanism with four links of different lengths, if the sum of the shortest link (S) and the longest link (L) is less than or equal to the sum of the other two links (P + Q), then at least one link will be capable of making a full rotation (360 degrees). Mathematically:

S + L ≤ P+ Q

Where:

• S = Length of the shortest link.
• L = Length of the longest link.
• P and Q = Lengths of the other two links.

If this condition is satisfied, the mechanism is called a Grashof mechanism.

Outcomes Based on Grashof’s Law:

1. Crank-Crank Mechanism (Double-Crank):
   • If S + L < P + Q, both adjacent links can rotate fully, resulting in a crank-crank mechanism (both cranks rotate 360 degrees).
2. Crank-Rocker Mechanism:
   • If S + L < P + Q and the shortest link is adjacent to the fixed link, one link will rotate fully (crank), while the other will oscillate (rocker), giving a crank-rocker mechanism.
3. Rocker-Rocker Mechanism (Double-Rocker):
   • If S + L > P + Q, no link can rotate fully. Instead, both links will oscillate, resulting in a rocker-rocker mechanism (both rockers swing back and forth).

Single Slider Crank Mechanism

A single slider crank mechanism is a mechanical system that converts rotary motion into linear motion, or vice versa. It consists of four main parts:

1. Crank: A rotating arm attached to the driving shaft.
2. Connecting Rod: A link that connects the crank to the slider.
3. Slider: The part that moves back and forth in a straight line.
4. Frame: The fixed part that holds everything together.