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# Mechanisms with Lower Pairs

Mechanisms with Lower Pairs

Contents

Recall the key terms: Transmission angle and Toggle positions 2

Define and classify Straight Line Mechanism 3

Recall Exact Straight Line Motion Mechanism 4

Describe Approximate Straight Line Motion Mechanism 5

Define Steering Gear Mechanism 6

Recall Davis Steering Gear 7

Describe Ackerman Steering Gear 8

Define Universal or Hooke’s Joint 9

Derive an expression for the Ratio of shafts velocities for Hooke’s Joint 10

Describe Maximum and Minimum speed of driven shaft 11

Describe the Condition of equal speeds 12

Recall Angular acceleration of driven shaft and Maximum fluctuation of speed 13

Define Double Hooke’s Joint 14

# Recall the key terms: Transmission angle and Toggle positions

Transmission angle and toggle positions are key terms used in the design and analysis of mechanisms. They are important factors that affect the performance and efficiency of a mechanism.

Transmission angle is the angle between the direction of the force being transmitted and the direction of motion of the output link. In other words, it is the angle between the force vector and the velocity vector of the output link. Transmission angle is an important factor in the design of mechanisms because it affects the efficiency and power transmission capacity of the system. Ideally, the transmission angle should be as close to 90 degrees as possible to maximise the efficiency of the system.

Toggle positions refer to the positions of the mechanism where the links are parallel to each other, and the mechanism becomes rigid. These positions are important because they determine the stability and rigidity of the mechanism. When the mechanism is in a toggle position, the links cannot move relative to each other, and the mechanism becomes rigid. This is useful in many applications where stability and precision are important.

Toggle positions are also important in the analysis of mechanisms because they affect the range of motion and the force transmission capacity of the system. In general, the more toggle positions a mechanism has, the more stable and rigid it will be. However, having too many toggle positions can also limit the range of motion of the mechanism and reduce its overall performance.

In summary, transmission angle and toggle positions are important concepts in the design and analysis of mechanisms. Transmission angle affects the efficiency and power transmission capacity of the system, while toggle positions determine the stability and rigidity of the mechanism. By understanding these key terms, engineers can design and analyze mechanisms that are efficient, stable, and high-performing.

# Define and classify Straight Line Mechanism

A straight line mechanism is a type of mechanism that is designed to produce a straight line motion output. The mechanism consists of rigid links connected by joints or pivots, which allow the links to move relative to each other in a specific way to produce the desired motion.

There are several types of straight line mechanisms, including:

1. Slider-crank mechanism: This is the most common type of straight line mechanism, consisting of a slider that moves along a straight line guided by a crankshaft that rotates. This mechanism is widely used in internal combustion engines to convert the reciprocating motion of the pistons into rotary motion.
2. Scotch yoke mechanism: This is a mechanism that converts rotary motion into linear motion using a sliding yoke that moves in a straight line. The yoke is driven by a rotating disc with an offset pin that slides inside the yoke, producing a straight line motion.
3. Whitworth quick return mechanism: This is a mechanism that produces a reciprocating straight line motion, with the return stroke being faster than the forward stroke. It consists of a rotating crank and a sliding block that is guided by a slot, producing a straight line motion.
4. Elliptical trammel mechanism: This mechanism produces a straight line motion by guiding a rod along a straight line while simultaneously moving it in a circular path. It is commonly used in drawing machines and engraving machines.
5. Pantograph mechanism: This mechanism is used for scaling or copying a drawing or image. It consists of four or more linked bars arranged in a parallelogram, with one end of the first bar connected to a stylus and the other end connected to a pivot point. The motion of the stylus is transmitted through the bars to produce a scaled copy of the original drawing.

In summary, a straight line mechanism is a type of mechanism that produces a straight line motion output. There are several types of straight line mechanisms, including the slider-crank mechanism, Scotch yoke mechanism, Whitworth quick return mechanism, elliptical trammel mechanism, and pantograph mechanism. Each of these mechanisms has specific characteristics and applications, and understanding their operation and design is important in the field of mechanical engineering.

# Recall Exact Straight Line Motion Mechanism

An Exact Straight Line Motion (ESL) mechanism is a type of straight line mechanism that is designed to produce an exact straight line output motion. Unlike other types of straight line mechanisms, the ESL mechanism can produce a straight line motion without any deviation, which is ideal for precision applications.

The most common types of ESL mechanisms are the Scott-Russell mechanism and the Hart mechanism.

The Scott-Russell mechanism consists of a series of four bars that are linked together by three pivots. One end of the first bar is fixed, and the other end is connected to the first pivot. The second bar is connected to the first pivot and the second pivot, and the third bar is connected to the second pivot and the third pivot. The fourth bar is connected to the third pivot and a sliding block, which moves along a straight line. When the first bar is rotated, the sliding block moves in a straight line without any deviation.

The Hart mechanism is similar to the Scott-Russell mechanism but uses a different linkage configuration. It consists of four bars that are linked together by four pivots. One end of the first bar is fixed, and the other end is connected to the first pivot. The second bar is connected to the first pivot and the second pivot, and the third bar is connected to the second pivot and the third pivot. The fourth bar is connected to the third pivot and a sliding block, which moves along a straight line. When the first bar is rotated, the sliding block moves in a straight line without any deviation.

In summary, an Exact Straight Line Motion (ESL) mechanism is a type of straight line mechanism that is designed to produce an exact straight line output motion. The most common types of ESL mechanisms are the Scott-Russell mechanism and the Hart mechanism, both of which use a linkage configuration to produce a straight line output motion without any deviation. These mechanisms are ideal for precision applications where a high degree of accuracy is required.

# Describe Approximate Straight Line Motion Mechanism

An Approximate Straight Line Motion (ASL) mechanism is a type of straight line mechanism that is designed to produce a straight line output motion that is close to an exact straight line motion, but not necessarily perfectly straight. Unlike the exact straight line motion mechanisms, ASL mechanisms can have some deviation in their output motion, but the deviation is kept within an acceptable range.

There are several types of ASL mechanisms, including the Watt’s mechanism, the Chebyshev straight-line mechanism, and the Peaucellier-Lipkin linkage.

The Watt’s mechanism is a four-bar linkage mechanism that is commonly used in steam engines to convert circular motion into linear motion. It consists of a fixed frame, a connecting rod, and two pivoting rods. The connecting rod is attached to two pivoting rods that are mounted on the frame. When the crankshaft is rotated, the connecting rod moves in a straight line, although the motion is not perfectly straight.

The Chebyshev straight-line mechanism is a six-bar linkage mechanism that is designed to produce a straight line output motion within a specific range of angles. It consists of a fixed frame, a connecting rod, and four pivoting links. When the crankshaft is rotated, the connecting rod moves in a straight line within a specified range of angles.

The Peaucellier-Lipkin linkage is a six-bar linkage mechanism that is also designed to produce a straight line output motion. It consists of a fixed frame, a pivoting input link, and four pivoting links. When the input link is pivoted, the output link moves in a straight line, although the motion is not perfectly straight.

In summary, an Approximate Straight Line Motion (ASL) mechanism is a type of straight line mechanism that is designed to produce a straight line output motion that is close to an exact straight line motion, but not necessarily perfectly straight. There are several types of ASL mechanisms, including the Watt’s mechanism, the Chebyshev straight-line mechanism, and the Peaucellier-Lipkin linkage, each of which uses a specific linkage configuration to produce a straight line output motion with some deviation within an acceptable range.

# Define Steering Gear Mechanism

Steering gear mechanism is a mechanical system used in vehicles to turn the front wheels and control their direction of movement. The steering gear mechanism is typically located in the steering column and consists of several interconnected parts that work together to translate the motion of the steering wheel into the turning of the front wheels.

The steering gear mechanism can be divided into two main types: manual and power steering. In a manual steering gear mechanism, the driver turns the steering wheel, which rotates a shaft that is connected to a series of gears. These gears, in turn, rotate the pitman arm and the drag link, which control the movement of the front wheels.

In a power steering gear mechanism, a hydraulic system is used to assist in the movement of the front wheels. This system typically consists of a power steering pump, which is driven by the engine, and a hydraulic cylinder, which is connected to the steering linkage. When the driver turns the steering wheel, the power steering pump generates pressure in the hydraulic cylinder, which makes it easier to turn the front wheels.

The most common type of steering gear mechanism is the rack and pinion system, which is used in many modern vehicles. In this system, the steering wheel is connected to a pinion gear, which rotates a rack gear. The rack gear is connected to the steering linkage and turns the front wheels. The rack and pinion system is known for its simplicity, durability, and precision.

In summary, the steering gear mechanism is a mechanical system used in vehicles to turn the front wheels and control their direction of movement. It can be manual or power-assisted, and is typically located in the steering column. The most common type of steering gear mechanism is the rack and pinion system, which is known for its simplicity, durability, and precision.

# Recall Davis Steering Gear

The Davis Steering Gear is a type of mechanical steering mechanism used in automobiles, particularly in vintage cars. It was invented by Joseph R. Davis in 1926 and patented in 1930. The Davis Steering Gear is a cam and lever mechanism that provides a smooth and precise steering control.

In the Davis Steering Gear, a lever is connected to the steering column, and a cam is connected to the front axle. When the driver turns the steering wheel, the lever moves the cam, which, in turn, moves the front wheels. The shape of the cam determines the steering ratio and the steering feel of the vehicle.

One of the key advantages of the Davis Steering Gear is its simplicity. It has fewer parts compared to other steering mechanisms, making it easier to maintain and repair. It is also known for its precision and smoothness, as the cam and lever mechanism provides a constant ratio and avoids any slack or play in the steering.

However, the Davis Steering Gear has some limitations. It can only be used on vehicles with a solid front axle, and it does not provide power assistance, making it difficult to turn the steering wheel at low speeds or when parking. Additionally, it is not as widely available as other steering mechanisms, which can make it challenging to find replacement parts.

In summary, the Davis Steering Gear is a type of mechanical steering mechanism used in automobiles, particularly in vintage cars. It is a cam and lever mechanism that provides a smooth and precise steering control, with fewer parts compared to other steering mechanisms. However, it has limitations such as the lack of power assistance and the difficulty in finding replacement parts.

# Describe Ackerman Steering Gear

The Ackerman steering gear is a type of mechanical steering mechanism used in automobiles that enables the front wheels to turn at different angles while maintaining a constant turning radius. This mechanism is named after its inventor, Rudolph Ackerman, who patented it in 1818.

In an Ackerman steering gear, the two front wheels are connected to a central pivoting point, which is located on the steering axis. When the driver turns the steering wheel, the wheels are turned at different angles, depending on their positions relative to the pivot point. The inside wheel turns at a sharper angle than the outside wheel, which allows both wheels to track along the same circular path, resulting in a constant turning radius.

The Ackerman steering gear is based on the principle that the steering angle of the wheels should be directly proportional to their distances from the pivot point. This ensures that both wheels maintain the same turning radius, which is essential for stable and safe handling.

One of the key advantages of the Ackerman steering gear is its ability to provide precise and stable steering. It is widely used in modern automobiles because it can be easily adapted to different wheelbases and turning radii, and it provides good steering response and control.

However, the Ackerman steering gear has some limitations. It can cause tire scrubbing, which can result in increased tire wear and reduced fuel efficiency. It can also be affected by variations in road surface and suspension, which can cause uneven tire wear and affect the handling and stability of the vehicle.

In summary, the Ackerman steering gear is a type of mechanical steering mechanism used in automobiles that enables the front wheels to turn at different angles while maintaining a constant turning radius. It is based on the principle that the steering angle of the wheels should be directly proportional to their distances from the pivot point. The Ackerman steering gear provides precise and stable steering but can cause tire scrubbing and be affected by variations in road surface and suspension.

# Define Universal or Hooke’s Joint

A Universal joint, also known as a Hooke’s joint, is a mechanical component that is used to connect two shafts that are inclined to each other. This joint is designed to transmit rotary motion between the two shafts without imparting any lateral forces.

A universal joint consists of a cross-shaped component that has four bearings at its corners. Each bearing sits inside a yoke or fork, which is attached to a shaft. The two yokes are connected by a cross-shaped component, which is designed to rotate around its center. The angle between the two shafts can be adjusted by changing the orientation of the yokes.

The universal joint is commonly used in applications where there is a need for transmitting rotary motion from a driving shaft to a driven shaft, where the two shafts are not parallel and there is a need for some flexibility in their connection. For example, in an automobile, the universal joint is used to connect the transmission shaft to the drive shaft, allowing the two shafts to be inclined to each other due to the movement of the suspension.

In addition to its use in automobiles, the universal joint is also commonly used in industrial machinery, marine vessels, and other applications that require the transmission of rotary motion between two non-parallel shafts.

# Derive an expression for the Ratio of shafts velocities for Hooke’s Joint

The Hooke’s joint, or universal joint, is a mechanism that is used to transmit rotary motion between two shafts that are inclined to each other. When the two shafts are inclined at an angle, the velocity of the driven shaft will not be the same as that of the driving shaft. The ratio of the shafts velocities can be calculated using the following expression:

Vd/Vr = [(cos α) / (cos β)]

where Vd is the velocity of the driven shaft, Vr is the velocity of the driving shaft, α is the angle between the driving shaft and the joint, and β is the angle between the driven shaft and the joint.

The angle α can be calculated using the following formula:

α = tan-1 [tan β / (cos ψ)]

where ψ is the angle between the two yokes of the universal joint.

By substituting the value of α in the first equation, we get:

Vd/Vr = [(cos tan-1 [tan β / (cos ψ)]) / (cos β)]

This expression can be simplified using the trigonometric identity:

cos tan-1 x = 1 / (1 + x2)0.5

After simplification, we get the final expression for the ratio of shafts velocities as:

Vd/Vr = [(1 + (tan2 β / (cos2 ψ)))0.5 / (cos β)]

This expression can be used to calculate the ratio of shaft velocities for any universal joint, given the angles between the shafts and the joint.

# Describe Maximum and Minimum speed of driven shaft

In a Hooke’s joint, the two shafts connected by the joint can rotate at different speeds due to the joint’s ability to accommodate angular misalignment between them. The speed of the driven shaft can be calculated based on the speed of the driving shaft and the angle of misalignment between the two shafts.

The maximum speed of the driven shaft occurs when the joint is operating at its maximum angle of misalignment. At this point, the driven shaft will be rotating at a maximum speed. The maximum speed of the driven shaft can be calculated using the following formula:

Vdmax = (Vd + Vr) / cosθ

Where Vdmax is the maximum speed of the driven shaft, Vd is the speed of the driving shaft, Vr is the radius of the driving shaft, and θ is the maximum angle of misalignment.

The minimum speed of the driven shaft occurs when the joint is operating at zero angle of misalignment. At this point, the driven shaft will be rotating at a minimum speed. The minimum speed of the driven shaft can be calculated using the following formula:

Vdmin = (Vd – Vr) / cosθ

Where Vdmin is the minimum speed of the driven shaft.

It is important to note that the maximum and minimum speeds of the driven shaft are dependent on the angle of misalignment between the two shafts. The joint should be designed to accommodate the maximum angle of misalignment that it will experience in operation to ensure that the driven shaft does not exceed its maximum speed.

# Describe the Condition of equal speeds

In a Hooke’s joint, the velocity ratio is the ratio of the angular velocities of the driving and driven shafts, and it depends on the angle of intersection between the two shafts. When the two shafts are at a small angle to each other, the velocity ratio is nearly constant and is equal to the cosine of the angle. However, as the angle increases, the velocity ratio becomes less constant and the driven shaft can experience significant speed fluctuations.

The maximum speed of the driven shaft occurs when the driving shaft is perpendicular to the driven shaft. In this condition, the velocity ratio is equal to zero, and the driven shaft is rotating at its maximum possible speed. The minimum speed of the driven shaft occurs when the driving shaft is at an angle of 45 degrees to the driven shaft. In this condition, the velocity ratio is equal to the cosine of 45 degrees, which is 0.707, and the driven shaft is rotating at 70.7% of its maximum speed.

In order to achieve equal speeds at the two shafts, the angle between the two shafts should be such that the velocity ratio is equal to one. This occurs when the angle is approximately 26.5 degrees. At this angle, the Hooke’s joint will provide a constant velocity ratio, and the driven shaft will rotate at the same speed as the driving shaft. However, it is important to note that this condition of equal speeds is only achievable for a limited range of angles and is not always possible in practical applications.

# Recall Angular acceleration of driven shaft and Maximum fluctuation of speed

The angular acceleration of the driven shaft in a Hooke’s joint is defined as the rate of change of its angular velocity with respect to time. The angular acceleration of the driven shaft is not constant, but varies as the joint rotates, causing a fluctuation in the speed of the driven shaft. The maximum fluctuation in speed occurs when the joint is at an angle of 45 degrees, which is the point of maximum angular acceleration.

The maximum fluctuation in speed is given by the equation:

ΔN = (N2 – N1) / (2 cos2(θ/2))

Where ΔN is the maximum fluctuation in speed, N1 is the speed of the driving shaft, N2 is the speed of the driven shaft, and θ is the angle through which the joint has turned.

If the angle θ is small, then cos2(θ/2) is close to 1, and the maximum fluctuation in speed is small. If θ is close to 90 degrees, then cos2(θ/2) is close to 0, and the maximum fluctuation in speed is large.

The condition of equal speeds occurs when the angular acceleration of the driven shaft is zero. This occurs when the joint is at an angle of 0 degrees or 180 degrees. At these angles, the speed of the driven shaft is equal to the speed of the driving shaft. However, the joint cannot be fixed at these angles because it must be able to rotate in order to transmit power between the driving and driven shafts. Therefore, the condition of equal speeds is only a theoretical condition.

# Define Double Hooke’s Joint

In mechanical engineering, a universal joint, also known as a Cardan joint, is a mechanical linkage that allows for the transmission of rotary motion and torque between two non-aligned shafts. It is commonly used in drivetrains that transmit power from an engine to wheels or other mechanisms. However, the universal joint introduces a non-uniform velocity ratio between the two shafts, resulting in fluctuations in speed and torque.

A Double Hooke’s Joint, also known as a double Cardan joint, is a mechanism used to overcome this problem. It is a modification of the basic universal joint and consists of two universal joints connected by a short shaft, with the second joint being connected to the driven shaft. This arrangement results in a constant velocity ratio between the two shafts and eliminates speed fluctuations.

Double Hooke’s Joint is commonly used in applications where the power transmission requires a joint capable of transmitting torque at a higher angle or where space limitations make it difficult to use a single universal joint. It is commonly used in automobiles to transmit power from the engine to the differential or axle, as well as in industrial applications such as cranes and conveyor systems.