ME301 MECHANICS OF MACHINERY

MODULE I

Introduction to kinematics and mechanisms - various mechanisms,

kinematic diagrams, degree of freedom- Grashof’s criterion, inversions,

coupler curves

Introduction to Mechanisms and Kinematics

Basic Definitions

• Machines are devices used to accomplish work. A mechanism is the heart of a machine. It is the mechanical portion of a machine that has the function of transferring motion and forces from a power source to an output.

• Mechanisms are assemblage of rigid members (links) connected together by joints (also referred to as Mechanical linkage or linkage).

o Links are the individual parts of the mechanism. They are considered rigid bodies and are connected with other links to transmit motion and force. Elastic parts, such as springs, are not rigid and, therefore, are not considered links. They have no effect on the kinematics of a mechanism and are usually ignored during kinematic analysis. They do supply forces and must be included during the dynamic force portion of analysis

Example

The adjustable height platform (on the right) is driven by hydraulic cylinders. Although the entire device could be called a machine, the parts that take the power from the cylinders and drive the raising and lowering of the platform are mechanisms.

o A joint is a movable connection between links and allows relative motion between the links. Some common joints

Joint Description

Allows pure rotation between the two links that it connects.

Revolute, pin or hinge

Allows linear sliding

between the links that it connects

Sliding, piston or prismatic

Permits two relative but dependent motions between the links being joined.

Screw joint

Permits the two links to slide and rotate relative to one another

Pin in a slot Joint

Allows for both rotation and sliding between the two links that it connects.

Cam Because of the complex motion permitted, the cam connection is called a higher-order joint, also called half join

A gear connection also allows rotation and sliding between two gears as their teeth mesh.

Gear

• A simple link is a rigid body that contains only two joints, which connect it to other links. A

complex link is a rigid body that contains more than two joints.

Kinematic Diagram

Kinematic analysis involves determination of position, displacement, rotation, speed, velocity, and acceleration of a mechanism. In analyzing the motion of a mechanism, it is often convenient to represent the parts in skeleton form (also referred to as kinematic diagram) so that only the dimensions that affect the motion are shown. A standard representation is used for the components of a mechanism as listed in the table.

Componen Typical form Kinematic representation t

Simple link

Simple link with point of interest

Complex link

Pin joint

Slider joint

Screw joint

Pin in a slot

joint

Cam joint

Gear Joint

• A kinematic diagram should be drawn to a scale proportional to the actual mechanism. For convenient reference, the links are numbered, starting with the frame, which serves as the frame of reference for the motion of all other parts. To avoid confusion, the joints should be lettered.

Examples o In the first diagram, the motion of the bar handle is a point of interest o In the second diagram, the motion of the end of the bottom jaw and the lower handle

are points of interest.

Typical form Kinematic representation

Mobility

The degree of freedom of a linkage is the number of independent inputs required to precisely position all links of the mechanism with respect to the frame. It can also be defined as the number of actuators needed to operate the mechanism. The number of degrees of freedom of a mechanism is also called the mobility, and it is given the symbol (M).

• When the configuration of a mechanism is completely defined by positioning one link, that

system has one degree of freedom.

Mobility in the planar case

A body moving freely in planar motion has three degrees of freedom (two translations on the plane and one rotation about an axis perpendicular to the plane). Suppose that in a given linkage there are (n) links.

• If they're all free to move independently, the system has Mobility (M = 3n). • If one link is chosen as the frame link, it is fixed to the base reference frame and loses all

its degrees of freedom. Therefore, the total Mobility of the system is (M = 3(n-1) ). • If a joint with connectivity fi (fi degrees of freedom of the joint) is formed between two bodies

the mobility of the system is diminished since those two bodies originally had three degrees of

freedom of motion relative to one another. After formation of the joint they have only fi

degrees of freedom of relative motion. Hence the reduction in the system mobility is 3 – fi.

Example

Without the pin joint, each link has 3 degrees of freedom. The pin joint however only allows free relative rotation (one degree of freedom) between the links. Relative translation is omitted (two free translations were omitted)

• If joints continue to be formed until there are j joints, the loss of system mobility is

�(3 − )

The total mobility of the linkage then becomes =1

= 3( − 1) − �(3 − ) =1

The above equation can be put in a simpler form also known as the Gruebler equation = 3( − 1) − 2 − ℎ

Where

is the total number of links in the mechanism

is the number of primary joints (pins or sliding joints) ℎ is the number of higher order joints (cam or gear joints)

Examples

Typical form Kinematic representation

n = 4 (four members), one of the members is frame, 4 pin joints with one degree of freedom = 3(4 − 1) − (3 − 1) − (3 − 1) − (3 − 1) − (3 − 1) = 3(4 − 1) − 4(2) = 1

Typical form Kinematic representation

n = 4 (four members), one of the members is frame, 3 pin joints with one degree of freedom

and one slider joint with one degree of freedom = 3(4 − 1) − (3 − 1) − (3 − 1) − (3 − 1) − (3 − 1) = 3(4 − 1) − 4(2) = 1

Typical form Kinematic representation

n = 3 (three members), one of the members is frame, 2 pin joints with one degree of freedom and one higher order joint

= 3(3 − 1) − (3 − 1) − (3 − 1) − (3 − 2) = 1

Notes: Because the mobility equation pays no attention to link size or shape, it can give misleading

results in the face of unique geometric configurations. For instance

Using the mobility equation we find = 3(5 − 1) − 6(3 − 1) = 0 Because the nodes

are equi-spaced and the three links are of equal length, the mobility of the mechanism is equal to 1 rather than 0

Assuming enough friction between the wheels to allow for rolling without slipping, the joint between the wheels has one degree of freedom. Using the mobility equation we get

= 3(3 − 1) − 3(3 − 1) = 0 The linkage however has one degree of freedom

When three links come together at a common pin, the joint must be modeled as two pins

= 3(6 − 1) − 6(3 − 1) − (3 − 1) = 1

Actuators and Drivers In order to operate a mechanism, an actuator, or driver device, is required to provide the input motion and energy. To precisely operate a mechanism, one driver is required for each degree of freedom exhibited. Many different actuators are used in industrial and commercial machines and mechanisms. Some of the more common ones

• Electric motors (AC) o Provide continuous rotary motion o Can only rotate at specific speed dependent on the frequency of the power line. o Single phase motors are available from 1/50 to 2 hp. Three phase motors are available

from 1/4 to 500 hp • Electric motors (DC)

o Provide continuous rotary motion o Require power from a generator or a battery o Allow control of speed and direction. o Can achieve speeds up to 30,000 rpm

• Engines o Provide continuous rotary motion o Rely on combustion of fuel, the speed can be anywhere from 1000 to 8000 rpm

• Servomotors o Coupled with a controller to produce a programmed motion. o The controller uses sensors to provide feedback information on the position, velocity,

and acceleration. o Characterized by lower power capacity than non-servomotors.

• Air or Hydraulic motors o Produce continuous rotary motion o Require compressed air or a hydraulic source

• Hydraulic or pneumatic cylinders o Produce limited linear stroke o Require air or hydraulic sources o The cylinder unit contains a rod and piston assembly that slides relative to the cylinder o The common kinematic representation for a hydraulic or pneumatic cylinder is

• Screw actuators o Produce limited linear stroke o Consist of a motor rotating a screw. A mating nut provides the linear motion o The kinematic diagram of screw actuators is similar to the cylinders

• Manual or hand operated actuator Actuators

Examples: For the outrigger in the figure below, the hydraulic cylinder is pinned to the foot to stabilize the

truck.

Mobility = 3 (4 − 1) – (3 − 1) – (3 − 1) – (3 − 1) – (3 − 1) = 1

= 3(5 − 1) − 4(3 − 1) − (3 − 2) − (3 − 1) = 1

The four-bar mechanism A four bar linkage is a mechanism made of four links, one being designed as the frame and connected by

four pins.

Example1: Four bar window wiper mechanism

Example2: Nose wheel mechanism

The link that is unable to move is referred to as the frame. Typically, the pivoted link that is connected to

the driver or power source is called the input link. The other pivoted link that is attached to the frame is designated the output link or follower. The coupler or connecting arm “couples” the motion of the

input link to the output link.

The mobility of these mechanisms is = 3(4 − 1) − 4(3 − 1) = 1

Because the four-bar mechanism has one degree of freedom, it is constrained or fully operated with one

driver. The wiper and Nose wheel systems above are activated by a single motor.

Grashof’s Criterion

Depending on the sizes of the links in the four-bar mechanism, the mechanism may act as one of the following

− Double crank where If one of the pivoted links is rotated continuously, the other pivoted link will also rotate continuously.

− Crank rocker where the output link will oscillate between limits. − Double rocker where both input and output links are constrained to oscillate between limits. − Change point where can be all the links can be positioned to becoming collinear. − Triple rocker where all three moving links rock

A criterion known as Grashof’s theorem can help differentiate among the different configurations

If S is the length of the shortest link

L the length of the longest link P and Q the lengths of the other two links

Then

• Grashof’s theorem states that a four-bar mechanism has at least one revolving link if: + ≤ +

• Conversely, the three non-fixed links will merely rock if: + > +

The results of Grashof’s theorem are summarized the table below

The double crank mechanism As specified in the criteria of Case 1 of the table, it has the shortest link of the four-bar mechanism configured as the frame. If one of the pivoted links is rotated continuously, the other pivoted link will also rotate continuously. Thus, the two pivoted links are both able to rotate through a full revolution. The double crank mechanism is also called a drag link mechanism.

Crank-Rocker As specified in the criteria of Case 2 of the table, it has the shortest link of the four-bar mechanism configured adjacent to the frame. If this shortest link is continuously rotated, the output link will oscillate between limits. Thus, the shortest link is called the crank, and the output link is called the rocker. The wiper system previously mentioned is designed to be a crank-rocker. As the motor continuously rotates the input link, the output link oscillates, or “rocks. ” The wiper arm and blade are firmly attached to the output link, oscillating the wiper across a windshield.

Double Rocker As specified in the criteria of Case 3 of the table, it has the link opposite the shortest link of the four-bar mechanism configured as the frame. In this configuration, neither link connected to the frame will be able to complete a full revolution. Thus, both input and output links are constrained to oscillate between limits, and are called rockers.

Change Point Mechanism As specified in the criteria of Case 4 of the table, the sum of two sides is the same as the sum of the other two. Having this equality, the change point mechanism can be positioned such that all the links become collinear.

Triple Rocker The triple rocker has no links that are able to complete a full revolution. Thus, all three moving links rock.

Other mechanisms

The Slider Crank Mechanism This mechanism consists of a combination of four links, with one being designated as the frame. The links are connected by three pin joints and one sliding joint. A mechanism that drives a manual water pump is an example. The corresponding kinematic diagram is also given.

The mobility of this mechanism is = 3(4 − 1) − 2 ∗ 3 − 2 ∗ 1 = 1

Because the slider-crank mechanism has one degree of freedom, it is constrained or fully operated with

one driver. The pump is activated manually by pushing on the handle.

Quick return mechanism Quick-return mechanisms exhibit a faster stroke in one direction than the other when driven at constant speed with a rotational actuator. They are commonly used on machine tools that require a slow cutting stroke and a fast return stroke.

crank-shaper linkage

offset slider-crank in a reciprocating saw

Scotch Yoke Mechanism A scotch yoke mechanism converts rotational motion to linear sliding motion, or vice Versa. A pin on a rotating link is engaged in the slot of a sliding yoke. With regards to the input and output motion, the scotch yoke is similar to a slider-crank, but the linear sliding motion is pure sinusoidal. In comparison to the slider-crank, the scotch yoke has the advantage of smaller size and fewer moving parts, but can experience rapid wear in the slot.

Geneva mechanism A Geneva mechanism converts the motion of a continuous rotating crank to a intermittent motion of the Geneva wheel. For a four-slot wheel, the Geneva wheel rotates 90 degrees for a complete rotation of a crank.

Linear Geneva As the crank rotates continuously, the output slider moves intermittently in the linear Geneva

mechanism.

Power Hacksaw This saw (in yellow) has reciprocating motion inside the arm (in green), driven by a rotating disk (in orange) with a cur (in grey) . This is a crank slider mechanism. The arm pivots at the center of the disk to feed the saw downward to cut the metal bar (in purple).

Oil field pump As the motor turns the crank (in orange), the walking beam (in yellow) oscillates. The pumping (sucker) rod, which is immersed in the oil, is connected to the horse -head of the walking beam by a cable. Therefore, the oscillation of the walking beam is converted to the reciprocating motion of the pumping rod to pump oil.

straight line mechanisms exact, approximate

Straight line motion mechanisms

Straight line motion mechanisms are mechanisms, having a point that moves along a straight line, or nearly along a straight line, without being guided by a plane surface.

Condition for exact straight line motion:

If point B (fig.1.40) moves on the circumference of a circle with center O and radius OA, then, point C, which is an extension of AB traces a straight line perpendicular to AO, provided product of AB and AC is constant.

Fig.1.40: Condition for exact straight line motion

Locus of pt.C will be a straight line, ┴ to AE if, is constant

Proof:

Peaucellier exact straight line motion mechanism:

Fig.1.41: Peaucellier exact straight line motion mechanism

Here, AE is the input link and point E moves along a circular path of radius AE = AB. Also, EC = ED = PC = PD and BC = BD. Point P of the mechanism moves along exact straight line, perpendicular to BA extended.

To prove B, E and P lie on same straight line:

Triangles BCD, ECD and PCD are all isosceles triangles having common base CD and apex points being B, E and P. Therefore points B, E and P always lie on the perpendicular bisector of CD. Hence these three points always lie on the same straight line.

To prove product of BE and BP is constant.

In triangles BFC and PFC,

and

But since BC and PC are constants, product of BP and BE is constant, which is the condition for exact straight line motion. Thus point P always moves along a straight line perpendicular to BA as shown in the fig.1.41.

Approximate straight line motion mechanism: A few four bar mechanisms with certain modifications provide approximate straight line motions.

Robert’s mechanism

Fig.1.42: Robert’s mechanism

This is a four bar mechanism, where, PCD is a single integral link. Also, dimensions AC, BD, CP and PD are all equal. Point P of the mechanism moves very nearly along line AB.

STEERING MECHANISMS

Hooke’s joint (Universal joints)

Hooke’s joins is used to connect two nonparallel but intersecting shafts. In its basic shape, it has two U –shaped yokes ‘a’ and ‘b’ and a center block or cross-shaped piece, C. (fig.1.47(a))

The universal joint can transmit power between two shafts intersecting at around 300 angles (α). However, the angular velocity ratio is not uniform during the cycle of operation. The amount of fluctuation depends on the angle (α) between the two shafts. For uniform transmission of motion, a pair of universal joints should be used (fig.1.47(b)). Intermediate shaft 3 connects input shaft 1 and output shaft 2 with two universal joints. The angle α between 1 and 2 is equal to angle α between 2 and 3. When shaft 1 has uniform rotation, shaft 3 varies in speed; however, this variation is compensated by the universal joint between shafts 2 and 3. One of the important applications of universal joint is in automobiles, where it is used to transmit power from engine to the wheel axle.

Fig.1.47(a)

Fig.1.47(b): Hooke’s joint

Steering gear mechanism

The steering mechanism is used in automobiles for changing the directions of the wheel axles with reference to the chassis, so as to move the automobile in the desired path. Usually, the two back wheels will have a common axis, which is fixed in direction with reference to the chassis and the steering is done by means of front wheels.

In automobiles, the front wheels are placed over the front axles (stub axles), which are pivoted at the points A & B as shown in the fig.1.48. When the vehicle takes a turn, the front wheels, along with the stub axles turn about the pivoted points. The back axle and the back wheels remain straight.

Always there should be absolute rolling contact between the wheels and the road surface. Any sliding motion will cause wear of tyres. When a vehicle is taking turn, absolute rolling motion of the wheels on the road surface is possible, only if all the wheels describe concentric circles. Therefore, the two front wheels must turn about the same instantaneous centre I which lies on the axis of the back wheel.

Condition for perfect steering

The condition for perfect steering is that all the four wheels must turn about the same instantaneous centre. While negotiating a curve, the inner wheel makes a larger turning angle θ than the angle φ subtended by the axis of the outer wheel.

In the fig.1.48, a = wheel track, L = wheel base, w = distance between the pivots of front axles.

Fig.1.48: Condition for perfect steering

From cotθ = and

from cotφ =

. This is the fundamental equation for correct steering. If this condition is satisfied, there will be no skidding of the wheels when the vehicle takes a turn.

Ackermann steering gear mechanism

Fig.1.49: Ackermann steering gear mechanism

fig.1.50: Ackermann steering gear mechanism

Ackerman steering mechanism, RSAB is a four bar chain as shown in fig.1.50. Links RA and SB which are equal in length are integral with the stub axles. These links are connected with each other through track rod AB. When the vehicle is in straight ahead position, links RA and SB make equal angles α with the center line of the vehicle. The dotted lines in fig.1.50 indicate the position of the mechanism when the vehicle is turning left.

Let AB=l, RA=SB=r; and in the turned position, . IE, the stub axles of inner and outer wheels turn by θ and φ angles respectively.

Neglecting the obliquity of the track rod in the turned position, the movements of A and B in the horizontal direction may be taken to be same (x).

Then, and

Adding, [1]

Angle α can be determined using the above equation. The values of θ and φ to be taken in this

equation are those found for correct steering using the equation . [2]

This mechanism gives correct steering in only three positions. One, when θ = 0 and other two each corresponding to the turn to right or left (at a fixed turning angle, as determined by equation [1]).

The correct values of φ, [φc] corresponding to different values of θ, for correct steering can be determined using equation [2]. For the given dimensions of the mechanism, actual values of φ, [φa] can be obtained for different values of θ. T he difference between φc and φawill be very small for small angles of θ, but the difference will be substantial, for larger values of θ. Such a difference will reduce the life of tyres because of greater wear on account of slipping.

But for larger values of θ, the automobile must take a sharp turn; hence is will be moving at a slow speed. At low speeds, wear of the tyres is less. Therefore, the greater difference between φc and φa larger values of θ ill not matter.

As this mechanism employs only turning pairs, friction and wear in the mechanism will be less. Hence its maintenance will be easier and is commonly employed in automobiles.

Geneva Mechanism: its history, function, and weaknesses

http://ebooks.library.cornell.edu/k/kmoddl/pdf/002_010.pdf

The Geneva mechanism is a timing device. According to Vector Mechanics for

Engineers for Ferdinand P. Beer and E. Russell Johnston Jr., says, “ [It] is used in

many counting instruments and in other applications where an intermittent rotary

motion is required.” (945) Essentially, the Geneva mechanism consists of a

rotating disk with a pin and another rotating disk with slots (usually four) into

which the pin slides (see right).

According to Brittanica.com, the Geneva mechanism

was originally invented by a watch maker. The

watch maker only put a limited number of slots in

one of the rotating disks so that the system could

only go through so many rotations. This prevented

the spring on the watch from being wound too tight,

thus giving the mechanism its other name, the

Geneva Stop. The Geneva Stop was incorporated

into many of the first film projectors used in

theaters.

In Optimum Design of Mechanical Elements, Ray C.

Johnson makes many references to the use of the Geneva mechanism to provide an

intermittent motion the conveyor belt of a "film recording marching." (13) He also

discusses several weak points in the Geneva mechanism. For instance, for each

rotation of the Geneva (slotted) gear the drive shaft must make one complete

rotation. Thus for very high speeds, the drive shaft may start to vibrate. Another

problem is wear, which is centralized at the drive pin. Finally, the designer has no

control over the acceleration the Geneva mechanism will produce.

Also, the Geneva mechanism will always

go through a small backlash, which stops

the slotted gear. This backlash prevents

controlled exact motion. (Picture at left

from Optimum Design of Mechanical Elements.)

Below are models of the Geneva

mechanism made with Working Model 2d

v4.0. The second model shows velocity

vectors for the slotted gear and the drive

shaft. Velocity is the black arrow and

acceleration is the green arrow. Move the

mouse over the mechanisms to start them

running.

MECHANICAL ADVANTAGE

À- � �p

As a mechanism moves over a range of motion its geometry changes. If we are using a

mechanisms to transmit torque, or force then we must consider the ratio between the input and

output force in various positions.

Transmission angle is the angle between the coupling member and the output member in a

mechanism. As this angle approaches ±90°, the mechanical advantage of the mechanism

typically increases.

Toggle positions occur when the input crank has near infinite mechanical advantage. Note: this

also applies that the follower has no mechanical advantage on the crank.

Consider the example below, [prob. 1-3 from Shigley & Uicker],

Displacement, Velocity and Acceleration Analysis of Plane Mechanisms

Velocity Analysis in Mechanism

Let a rigid link OA, of length r rotate about a fixed point O with a uniform angular velocity

rad/s in a counter-clockwise direction OA turns through a small angle δθ in a small

interval of time δt. Then, A will travel along the arcAA’ as shown in figure.

∴ Velocity of A relative to O

∴ In the limits, when

Thus, velocity of A is ωr and is perpendicular to OA.

Velocity of Intermediate Point

If represent the velocity of B with respect to O, then

i.e., b divides the velocity vector in the same ratio as B divides the link. The magnitude of the

linear velocity of a point on the rotating body at a particular instant is proportional to its distance

fromt the axis of rotation.

Velocity Images of Four Link Mechanism

Figure shows a four link mechanism (quadric cycle mechanism) ABCD in which AD is

fixed link and BC is the coupler. AB is the driver rotating at an angular speed of ω rad/s in

the clockwise direction if it is a crank or moving at angular velocity ω at this instant if it

is rocker.

Velocity Images of Slider-Crank Mechanism

Consider a slider-crank mechanism in which OA is the crank moving with uniform angular

velocity ω rad/s in the clockwise direction. At point B, a slider moves on the fixed

guide G.

From the given configuration, the coupler AB has angular velocity in the counter-clockwise

direction. The magnitude being .

Velocity of Rubbing

Let us take two links of a turning pair, a pin is fixed to one of the links whereas a hole is

provided in the other to fit the pin. When joined the surface of the hole of one link will rub

on the surface of pin of the other link. The velocity of rubbing of the two surfaces will

depend upon the angular velocity of a link relative to the other.

Pin at A

The pin at A joins links AD and AB. AD being fixed, the velocity of rubbing will depend upon the angular velocity of AB only.

Velocity of rubbing = raω

where, ra = radius of pin at A

Pin at B

ωba = ωab = ω (clockwise)

(counter-clockwise)

rb = Radius of pin at B

Velocity of rubbing = rb(ωab + ωbc)

Pin at C

ωbc = ωcb (counter-clockwise)

ωdc = ωcd (clockwise)

rc = Radius of pin at C

Velocity of rubbing = rc(ωbc + ωdc)

Pin at D

where, rd = radius of pin at D

Velocity of rubbing = rdωcd

Instantaneous Centre of Velocity (I-centre)

The instantaneous centre of velocity can be defined as a point which has no velocity with

respect to the fixed link.

Suppose there are two link 1 and link 2

Link 1 may not be fixed. Rigid body 2 is shown to be in plane motion with respect to the

link 1.

In case of fixed link, (link 2) velocity of the point A and B are proportional

toPA and PB respectively. So, instantaneously, the rigid body can be thought of as being

momentarily in pure rotation about the point P. The velocity of any point C on the body at

this instant is given by in a direction perpendicular to PC. This point P is

called the instantaneously centre of velocity and its instantaneously velocity is zero.

If both links 1 and 2 are in motion, we can define a relative instantaneous centre P 12 to be

a point on 2 having zero relative velocity with respect to a coincident point on 1.

Consequently, the relative motion of 2 with respect to 1 be appears to be pure rotation

about P12. So P21 and P12 are identical.

Centro

Instantaneous centre is also called centro. So, two coincident points belonging to two

bodies having relative motion with the properties.

They have the same velocities.

They form a point in one of the rigid bodies about which the other rotates and vice-versa.

Which is perhaps true for only an instant.

Primary Centro One which can be easily located by a mere observation of the mechanism.

Secondary Centro Centros that cannot be easily located.

Instantaneous Centre of Acceleration

It is defined as a point on a link having zero relative acceleration with resp ect to a

coincident point on the other link and is different from the instantaneous centre of

velocity.

Aronhold-Kennedy Theorem of Three Centre

It state that if three bodies are in relative motion with respect to one another, the three

relative instantaneous centers of velocity ar collinear.

P12 - Instantaneous centre of fixed ground 1 and body 2.

P13 - Instantaneous centre of fixed ground 1 and body 3.

P23 - Instantaneous centre of body 2 and body 3.

Number of Centros in a Mechanism

For a mechanism of n links, the number of centros (Instantaneous centre) Nis

Number of Lines of Centros

The number of lines of centros L for a mechanism with n links is

Acceleration Analysis in Mechanism

The rate of change of velocity with respect to time is known as acceleration and acts in

the direction of the change in velocity. Velocity can changed by only changing its

magnitude or its direction. Let a link OA, of length r, rotate in a circular path in the

clockwise direction as shown in figure. It has an instantaneously angula r velocity ω and

an angular acceleration α in the same direction i.e., the angular velocity increases in the

clockwise direction.

Tangential acceleration of A relative O is defined as

Centripetal or radial acceleration of A relative to O is defined as

Total acceleration (net acceleration)

There are three cases occurred in the net acceleration as given below

Case I When α = 0 ⇒ ω = constant

So, net acceleration

Case II When ω = 0 ⇒ A has linear motion as

Net acceleration

Case III: When α is negative or the link OA decelerates, tangential acceleration will be negative or

its direction will be as shown in figure.

Corial’s Acceleration Component

Consider a link AR rotates about a fixed point A on it. P is a point on a slider on the link.

Here, ω = Angular velocity of the link

α = Angular acceleration of the link

v = Linear velocity of the slider on the link

f =Linear acceleration of the slider on the link

r = Radial distance of point P on the slider.

Key Points

Direction of coriol’s acceleration component (2ω.v) is perpendicular to AR Coriol’s component is

positive if

The link AR rotate clockwise and the slider moves radially outward.

The link rotate counter clockwise and the slider moves radially inwards. Acceleration of

slider (f) is positive if

Slider has a deceleration while moving in the inward direction.

Slider has acceleration while moving in the outward direction.

Acceleration of P | | to AR = acceleration of slider – centripetal

acceleration

Acceleration of P ⊥ to AR. = Coriol’s acceleration + tangential

acceleration

Let Q be a point on the link AR immediately beneath the point P at the instant, then

Acceleration of P = acceleration of P | | to AR + acceleration of P ⊥ to AR

= acceleration of P relative to Q + Acceleration of Q relative to A + Coriols acceleration

component

Instant centre -Kennedy’s theorem

The transformation of motion from one shaft to another shaft requires three bodies: a frame

(the position of each shaft is fixed in the frame) and two gears. Consider a general case

when two disks with arbitrary profiles shown in figure 1, represent gears 2 and 3. Also

assume that disk 2 rotates with the constant angular velocity ω2. The motion is transferred

through the direct contact at point P (note that P2 and P3 are the same point P, but the first

point is associated with disk 2 whereas the second point is associated with disk 3). We

need to find out whether or not the angular velocity ω3 will remain constant and if not, what

is required to make it remain constant. The answer is given by Kennedy’s theorem.

Kennedy’s theorem identifies the fundamental property of three rigid bodies in motion.

First, recall about that the instantaneous center of velocity, it is defined as the

instantaneous location of a pair of coincident points of two different rigid bodies for

which the absolute velocities of two points are equal. If one considers body 2 and the

frame (represented by point O2) in figure 1, then the instantaneous center of these

two bodies is point O2, which belongs to the frame and to disk 2. The absolute velocities of

both bodies at point O2 are zero. The same is valid for disk 3 and the frame represented by

point O3. For the three bodies in motion there are three instantaneous centers: for all

combinations of pairs. Thus, there is an instantaneous center between the two disks.

Figure 1: Kennedy’s theorem illustration

Since point P is a common point for two disks in figure 1, for each disk the velocity

component at this point directed along the common normal is the same and equal to Vp.

One can move this velocity vector along the common normal until it intersects the line

connecting the two instantaneous centers O2 and O3 at point C. According to Kennedy’s

theorem, this point (P) is the instantaneous center of velocity for the two disks. Indeed, the

velocity Vp, the only instantaneous common velocity for the two disks (disk 2, disk 3) will be

equal to

Triangles AO2C and BO3C are similar, it follows that

From the above relationship, the ratio of angular velocities is equal to

Where lengths l2 and l3 denote O2C and O3C, respectively (see the above figure 1).

Velocity Vc in figure 1 is a common velocity for the two disks since

Figure 2: Illustration of the involute profile generation

Velocity Vc is also the absolute velocity of each body at this point because there cannot be

another velocity component along the line O2O3 (since the distance between the frame

points O2 and O3 is not changing and bodies 2 and 3 are assumed to be rigid).

Note that the relative velocity of two disks at point C is zero, whereas relative velocity at

point P is not zero.

The above equation 3 gives the transformation of angular velocities from disk 2 to disk 3. It

can be seen from the above relation, in order to remain kinematic ratio (ω2/ω3) constant,

legthss l2 and l3 must not change. It is clear from figure 1, however, that for arbitrary profile

shapes the common normal changes its direction during the motion, and thus point C

moves along line O2O3. Thus, the problem of meeting the constant ratio requirement is to

find such disk profiles that the kinematic ratio remains constant. It will be shown in the

following that if the disk profile is described by the involute function, the common normal

does not change its direction.