Velocity and Acceleration in Mechanisms

Contents

Define Instantaneous Centre of rotation 2

Define the key points such as centrode, axode, types of instantaneous centres etc 3

State the Kennedy’s theorem 4

Recall the method to find and locate the Instantaneous centres in mechanism 4

State the theorem of Angular velocity ratio 5

Recall the concept of Relative motion 6

Draw the velocity diagram through Relative velocity method 7

Describe the Rubbing velocity of a pin-joint 9

Recall the term Mechanical advantage 10

Recall the accelerations in circular motion 11

Describe the Acceleration diagram for a link 12

Recall the acceleration of a point on a link 13

Describe Coriolis component of acceleration 14

Describe Klein’s construction for a Single slider crank mechanism 15

Define Instantaneous Centre of rotation

The instantaneous center of rotation (ICR) is a theoretical point in a mechanism or body that has zero velocity at any given instant. It is the point about which a body or a mechanism appears to rotate at a particular instant, even though there may be no actual physical rotation happening at that point.

The concept of the instantaneous center of rotation is widely used in kinematics and dynamics of machines and mechanisms. It is used to determine the direction and magnitude of the velocity and acceleration of various points in a mechanism, and to study the motion of the mechanism as a whole.

The ICR can be found by analyzing the motion of a body or mechanism at any given instant. It is the point where the velocities of two points in the body or mechanism are equal and opposite, and are perpendicular to the line joining those two points. The ICR can be located at any point on this line, and its position will change as the motion of the mechanism changes.

The concept of the ICR is particularly useful in designing and analyzing machines and mechanisms, as it allows engineers to study the motion of the mechanism at any given instant, and to design mechanisms that have desirable properties, such as low vibration, low wear, and high efficiency. The ICR can also be used to analyze and optimize the motion of various parts in a mechanism, and to improve the overall performance of the mechanism.

Overall, the instantaneous center of rotation is a critical concept in kinematics and dynamics of machines and mechanisms. It allows engineers to study and understand the complex motion of mechanisms and to design mechanisms that have desirable properties and high efficiency.

Define the key points such as centrode, axode, types of instantaneous centres etc

When studying the motion of machines and mechanisms, several key points and concepts are often used to understand their behavior. These include:

1. Centrode: The centrode is a geometric curve that describes the path of the instantaneous centres of rotation of a mechanism as it moves through its motion. It is an important concept as it provides insight into the motion and forces acting on a mechanism.
2. Axode: The axode is the locus of the centres of rotation of one link of a mechanism relative to another as it moves through its motion. The axode is a key concept in the design and analysis of mechanisms as it provides insight into the motion and forces acting on the links.
3. Instantaneous center of rotation: The instantaneous center of rotation (ICR) is a theoretical point in a mechanism or body that has zero velocity at any given instant. It is the point about which a body or a mechanism appears to rotate at a particular instant, even though there may be no actual physical rotation happening at that point.
4. Types of instantaneous centres: There are three types of instantaneous centres – sliding, turning, and fixed. Sliding centres are those that slide along a path, turning centres are those that rotate about a fixed axis, and fixed centres are those that remain stationary.
5. Kennedy’s theorem: Kennedy’s theorem states that the centroids of two gears in mesh will intersect at a point called the pitch point. This theorem is useful in the design of gear trains and the calculation of gear ratios.

Overall, understanding these key points and concepts is important in the design and analysis of machines and mechanisms. They allow engineers to study the motion and forces acting on a mechanism, and to optimize its performance.

State the Kennedy’s theorem

Kennedy’s theorem is a fundamental concept in the theory of gearing that relates the motion and geometry of two meshing gears. The theorem states that the instantaneous centres of rotation of the two gears, which lie on their respective pitch circles, will always intersect at a single point called the pitch point. The pitch point is the point at which the two gears are in contact, and it is the only point on the pitch circles where the velocity of both gears is equal in magnitude and opposite in direction.

The pitch point is a critical concept in the design of gears and gear trains. The pitch circle diameter is used to determine the gear ratio, and the location of the pitch point determines the relative speed and direction of rotation of the two gears. By analyzing the motion and forces at the pitch point, engineers can optimize the design of gear systems to improve their efficiency, durability, and performance.

Overall, Kennedy’s theorem is a fundamental concept in the theory of gearing, and it provides a powerful tool for the design and analysis of gear systems.

Recall the method to find and locate the Instantaneous centres in mechanism

The instantaneous center is a critical concept in the analysis of mechanisms, particularly in the determination of their velocities and accelerations. There are several methods to find and locate instantaneous centres in a mechanism:

1. Velocity Polygon Method: This method involves constructing the velocity polygons for two points on the moving links of a mechanism. The intersection of the lines drawn through the corresponding sides of the two polygons is the instantaneous center of rotation.
2. Instantaneous Center of Zero Velocity (ICZV) Method: This method involves identifying a point on a moving link that has zero velocity. The perpendicular bisector of the line connecting this point to the corresponding point on the other moving link will pass through the instantaneous center of rotation.
3. Intersection of Tangents Method: This method involves drawing tangents to the moving links of a mechanism at two different points. The instantaneous center of rotation is located at the intersection of these two tangents.
4. Rolling Contact Method: This method is used for gears and other rolling elements that have a rolling contact. The instantaneous center of rotation is located at the point where the two rolling elements are in contact.
5. Centrode Method: The centrode is the locus of the instantaneous centres of a mechanism. The instantaneous centres can be located by finding the intersections of the centrode with the moving links of the mechanism.

Overall, the above-mentioned methods can be used to find and locate instantaneous centres in a mechanism. The selection of a particular method depends on the complexity of the mechanism and the required accuracy of the analysis.

State the theorem of Angular velocity ratio

The theorem of angular velocity ratio is an important principle in the study of mechanisms. It states that the ratio of the angular velocities of any two rotating links in a mechanism is equal to the ratio of the lengths of the perpendiculars drawn from their respective instantaneous centres to the line of centres of the two links.

This means that if two links in a mechanism are rotating about a common fixed center, the angular velocity of one link is directly proportional to the length of the perpendicular from its instantaneous center to the line of centres, and inversely proportional to the length of the perpendicular from the instantaneous center of the other link to the line of centres.

The angular velocity ratio theorem is particularly useful in the design and analysis of gear trains, where it can be used to determine the gear ratios required to achieve a desired speed output. By adjusting the sizes of the gears in a train, the angular velocities of the different links can be controlled to achieve the desired output speed and torque.

Overall, the theorem of angular velocity ratio is a fundamental principle in the study of mechanisms and is widely used in the analysis and design of various types of machinery.

Recall the concept of Relative motion

The concept of relative motion is an important concept in the study of mechanics and is used to describe the motion of one object or point with respect to another object or point.

In simple terms, relative motion refers to the motion of an object or point as observed from the point of view of another object or point that is itself in motion. For example, if two cars are driving down a road, each car is in motion relative to the ground, but they are also in motion relative to each other. From the point of view of the driver of one car, the other car appears to be moving in a certain direction and at a certain speed, while from the point of view of the driver of the other car, the first car appears to be moving in the opposite direction and at a different speed.

Relative motion is an important concept in the analysis of mechanical systems, as it can be used to describe the motion of one part of a system with respect to another part. By understanding the relative motion of different parts of a system, engineers can design mechanisms that perform specific tasks, such as converting rotary motion into linear motion or vice versa.

Overall, the concept of relative motion is a fundamental concept in the study of mechanics and is used extensively in the analysis and design of various types of mechanical systems.

Draw the velocity diagram through Relative velocity method

In kinematics, the velocity diagram is a graphical representation of the velocities of different points in a mechanism. It is used to analyze the relative motion between different components of the mechanism. There are several methods to draw a velocity diagram, one of which is the relative velocity method.

The relative velocity method involves finding the relative velocities of different points in a mechanism by considering their motions relative to each other. To draw the velocity diagram using the relative velocity method, the following steps can be followed:

1. Identify the points whose velocities need to be determined. These points are usually the key points in the mechanism, such as the input and output points.
2. Draw a free-body diagram of the mechanism, showing all the forces and moments acting on each component.
3. Identify the axes of rotation for each component of the mechanism.
4. Use the relative velocity equation, which states that the velocity of one point with respect to another is equal to the sum of their velocities, to determine the relative velocities of the different points in the mechanism.
5. Once the relative velocities have been determined, they can be plotted on the velocity diagram. The length and direction of each velocity vector represent the magnitude and direction of the velocity at the corresponding point in the mechanism.

Finally, the velocity diagram can be used to analyze the motion of the mechanism and to determine important parameters such as the velocity ratio, acceleration, and direction of motion.

Overall, the relative velocity method is a powerful tool for analyzing the motion of mechanisms and for designing new mechanisms with specific performance characteristics. By understanding the relative velocities of different points in a mechanism, engineers can optimize the design for maximum efficiency and performance.

Describe the Rubbing velocity of a pin-joint

In a mechanism, a pin-joint is a type of joint that allows two or more components to rotate relative to each other around a common axis. However, due to the manufacturing and assembly tolerances, there can be small deviations in the location of the joint centres, which can result in the phenomenon of rubbing or sliding between the mating surfaces of the components.

Rubbing velocity is the velocity of the sliding or rubbing motion between the two mating surfaces of a pin-joint. It is caused due to the non-ideal conditions in a mechanism where the center of rotation of the joint does not coincide with the geometrical center of the joint. As a result, the joint rotates about an axis that is slightly offset from the ideal axis, leading to a small amount of sliding or rubbing between the mating surfaces.

Rubbing velocity can be expressed in terms of the relative angular velocity between the components on either side of the joint and the perpendicular distance between the actual center of rotation and the ideal center of rotation. The rubbing velocity is a vector quantity that has both magnitude and direction, and it is proportional to the relative angular velocity of the components on either side of the joint.

Rubbing velocity can have several undesirable effects on the performance of a mechanism. It can cause wear and tear on the mating surfaces of the components, leading to reduced lifespan and performance. It can also introduce unwanted vibrations and noise in the mechanism, leading to decreased efficiency and accuracy. Therefore, it is important to minimize the rubbing velocity of a pin-joint in order to optimize the performance of a mechanism.

Overall, the rubbing velocity of a pin-joint is an important concept in the study of kinematics and is essential for designing efficient and reliable mechanical systems. By understanding the factors that contribute to rubbing velocity, engineers can develop strategies to minimize its effects and improve the performance of their designs.

Recall the term Mechanical advantage

The term mechanical advantage refers to the ratio of the output force produced by a machine to the input force applied to it. It is a measure of the amplification of force achieved by using a machine. Machines are designed to make it easier for us to do work by allowing us to apply a smaller force over a greater distance. This can be accomplished by using a lever, a pulley, a wheel and axle, or some other type of machine.

The mechanical advantage of a machine is determined by the ratio of the output force to the input force. For example, if a machine produces an output force of 100 N and the input force applied to the machine is 20 N, then the mechanical advantage of the machine is 5.

Mechanical advantage can be expressed in different ways depending on the type of machine being used. For example, in the case of a lever, mechanical advantage is the ratio of the length of the lever arm on the output side of the fulcrum to the length of the lever arm on the input side of the fulcrum. In the case of a pulley, mechanical advantage is the number of supporting strands of rope or cable attached to the load being lifted.

The concept of mechanical advantage is important in machine design because it allows engineers to design machines that require less force to do the same amount of work, making them more efficient and effective.

Recall the accelerations in circular motion

In circular motion, an object moves along a circular path. The velocity of the object constantly changes direction, and this change in velocity is called acceleration. There are two types of acceleration in circular motion: tangential acceleration and radial acceleration.

Tangential acceleration is the acceleration of an object moving in a circular path in the tangential direction. This type of acceleration is caused by a change in the speed of the object. When an object moves in a circle at a constant speed, it still experiences tangential acceleration because its velocity is constantly changing direction.

Radial acceleration, on the other hand, is the acceleration of an object moving in a circular path in the radial direction. This type of acceleration is caused by a change in the direction of the velocity vector. As the object moves along the circular path, its velocity vector is constantly changing direction, which results in a radial acceleration.

The total acceleration of an object moving in circular motion is the vector sum of its tangential and radial accelerations. The direction of the total acceleration is always towards the center of the circular path, and its magnitude is given by the following equation:

a = √(at² + ar²)

where a is the total acceleration, at is the tangential acceleration, and ar is the radial acceleration.

Describe the Acceleration diagram for a link

The acceleration diagram for a link in a mechanism is a graphical representation of the magnitude and direction of acceleration of any point on the link. It is useful in analyzing the kinematics of a mechanism to determine the velocity and acceleration of different parts of the mechanism.

To construct the acceleration diagram for a link, first, draw the position diagram for the link, which represents the positions of the link at different instants of time during its motion. Next, draw the velocity diagram for the link, which represents the magnitude and direction of velocity of any point on the link at different instants of time. Finally, draw the acceleration diagram for the link, which represents the magnitude and direction of acceleration of any point on the link at different instants of time.

The acceleration of a point on a link can be resolved into two components, tangential and normal acceleration. The tangential acceleration is the component of acceleration along the tangent to the path of motion of the point, while the normal acceleration is the component of acceleration perpendicular to the tangent to the path of motion. The magnitude and direction of the tangential and normal acceleration can be determined from the velocity and acceleration diagrams.

The acceleration diagram is particularly useful in determining the forces acting on a link, as the forces are directly proportional to the acceleration. By analyzing the acceleration diagram, it is possible to determine the magnitude and direction of the forces acting on a link, which can be used to design and optimize the mechanism for maximum efficiency and performance.

Recall the acceleration of a point on a link

In the study of mechanisms, it is important to analyze the motion of individual points on the links to understand the overall motion of the mechanism. One of the key factors to consider is the acceleration of a point on a link, which is a vector quantity that describes the rate of change of the point’s velocity over time.

The acceleration of a point on a link can be broken down into two components: tangential acceleration and normal acceleration. Tangential acceleration is the component of acceleration that is in the direction of the velocity vector of the point, and it represents the rate of change of the point’s speed. Normal acceleration is the component of acceleration that is perpendicular to the velocity vector of the point, and it represents the rate of change of the direction of the point’s velocity.

To calculate the acceleration of a point on a link, we need to know the velocity of the point and the rate of change of the point’s velocity, which is called its angular acceleration. The angular acceleration is related to the angular velocity of the link and its time derivative, the angular acceleration.

The acceleration diagram is a graphical representation of the tangential and normal acceleration vectors at a point on a link. The tangential acceleration vector is drawn tangential to the path of motion of the point, and the normal acceleration vector is drawn perpendicular to the tangential acceleration vector, in the direction of the center of curvature of the path. The acceleration diagram can be used to study the variation of acceleration over the course of a link’s motion and to identify the maximum and minimum acceleration values.

Overall, the analysis of the acceleration of a point on a link is an important aspect of the study of mechanisms, as it allows for a deeper understanding of the motion of the mechanism and the forces involved in that motion.

Describe Coriolis component of acceleration

In kinematics, the Coriolis component of acceleration refers to an acceleration experienced by a point on a moving link due to the angular velocity and angular acceleration of the link. This component is perpendicular to the velocity of the point and tangential to the circular path followed by the point.

The Coriolis component of acceleration can be expressed as:

a C = -2v x ω’ – ω x (ω x r)

where a_C is the Coriolis component of acceleration, v is the velocity of the point, ω is the angular velocity of the link, ω’ is the angular acceleration of the link, r is the position vector of the point relative to the instantaneous center of rotation, and x denotes the vector cross product.

The first term on the right-hand side of the equation represents the acceleration caused by the angular acceleration of the link, while the second term represents the acceleration caused by the angular velocity of the link. The Coriolis component of acceleration is important in the analysis of mechanisms that involve moving links, such as robot arms or cam-follower systems.

In practical terms, the Coriolis component of acceleration can have an effect on the performance of a mechanism, particularly when the mechanism is required to move at high speeds or when high precision is required. It can lead to vibrations or other undesired movements in the system, which may need to be minimised through careful design or the use of specialised components such as dampers or bearings.

Describe Klein’s construction for a Single slider crank mechanism

Klein’s construction is a graphical method used to determine the velocity and acceleration of the slider in a single slider crank mechanism. This method was developed by Jacob Klein, a German mathematician in the 19th century.

To perform the Klein’s construction, the following steps are taken:

1. Draw the crank and connecting rod to scale with the given dimensions.
2. Draw a vertical line to represent the slider.
3. Draw the perpendicular bisectors of the crank and the connecting rod.
4. Draw a circle with a radius equal to the length of the connecting rod, centred at the crank pin.
5. Draw another circle with a radius equal to the length of the crank, centred at the crank pin.
6. Mark the intersection points of the circle with the vertical line drawn earlier.
7. Join these intersection points to the center of the crankshaft to form two radial lines.
8. The velocity of the slider at any point can be found by projecting the intersection point on the vertical line to the crank axis and measuring the velocity of the corresponding point on the crank circle.
9. The acceleration of the slider can be found by the second construction, which involves drawing an arc of a circle with a radius equal to the length of the crank at the point where the vertical line intersects with the crank circle. The tangent to this arc at the intersection point with the vertical line gives the acceleration of the slider at that point.

Klein’s construction is a useful tool to visually understand the motion of a single slider crank mechanism and can be used to determine the maximum velocity and acceleration of the slider, which are important considerations in the design of machinery.