Lami's Theorem

In statics, Lami's theorem is an equation that relates the magnitudes of three coplanar, concurrent and non-collinear forces, that keeps a body in static equilibrium.

Lami’s theorem states that if three forces acting at a point are in equilibrium, each force is proportional to the sine of the angle between the other two forces.

Consider three forces A, B, C acting on a particle or rigid body making angles α, β and γ with each other.

According to Lami’s theorem , the particle shall be in equilibrium if

The angle between the force vectors is taken when all the three vectors are emerging from the particle.

Belt Drives

A belt is a looped strip of flexible material used to mechanically link two or more rotating shafts. A belt drive offers smooth transmission of power between shafts at considerable distance. Belt drives are used as source of motion to transfer to efficiently transmit power or to track relative movement.

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Types of Belt Drives:

In a two pulley system, depending upon direction the belt drives the pulley, the belt drives are divided into two types. They are open belt drive and crossed belt drive. The two types of belt drives are discussed below in brief.

Open belt drives :

An open belt drive is used to rotate the driven pulley in the same direction of driving pulley.  In motion of belt drive, power transmission results makes one side of pulley more tightened compared to the other side.  In horizontal drives, tightened side is always kept in the lower side of two pulleys because the sag of the upper side slightly increases the angle of folding of the belt on the two pulleys.

Crossed belt drives :

A crossed belt drive is used to rotate driven pulley in the opposite direction of driving pulley. Higher the value of wrap enables more power can be transmitted than an open belt drive. However, bending and wear of the belt are important concerns.

Advantages of belt drives :

• Belt drives are simple are economical.
• They don't require Parallel shafts.
• Belts drives are provided with overload and jam protection.
• Noise and vibration are damped out. Machinery life is increased because load fluctuations are shock-absorbed.
• They are lubrication-free. They require less maintenance cost.
• Belt drives are highly efficient in use (up to 98%, usually 95%).
• They are very economical, when distance between shafts is very large.

Disadvantages of belt drives :

• In Belt drives, angular velocity ratio is not necessarily constant or equal to the ratio of pulley diameters, because of slipping and stretching.
• Heat buildup occurs. Speed is limited to usually 35 meters per second. Power transmission is limited to 370 kilowatts.
• Operating temperatures are usually restricted to –35 to 85°C.
• Some adjustment of center distance or use of an idler pulley is necessary for wearing and stretching of belt drive compensation. 

Newton's Laws of Motion

Newton’s First Law of Motion :

Newton’s first law of motion states that every object will remain at rest or in uniform  motion in a straight line unless compelled to its state by the action of an external force.

The first law of motion is normally taken as the definition of inertia. If there is no net force acting on an object then object will remain a constant velocity. If an external force is applied, the velocity of body will change because of force.

Newton’s Second law of Motion :

Newton’s Second law of motion states that if the resultant force acting on a particle is not zero, the particle will have acceleration proportional to the magnitude of the resultant and in the direction of this resultant force.  This law explains how velocity of an object changes when it is subjected to an external force. The law defines force to be equal to change in momentum (mass times velocity) per unit time.

For an object  with constant mass m, Newton’s second law of motion states that the force 'F' is the product of an object’s mass 'm' and its acceleration 'a'.

F = m.a

For an externally applied force, the acceleration depends on mass of the object and a change in velocity will generate a force. The above equation works in both ways.

Newton’s Third Law of Motion :

Newton’s third law of motion states that for every action (force) in nature there is an equal and opposite reaction. In other words, if object 'A' exerts a force on object 'B', then object 'B' also exerts an equal force on object 'A'.

The third law of motion can be used to explain the generation of lift by a wing and the production of thrust by a jet engine.

Heat Exchangers classification

Heat exchanger is a process equipment designed for the effective transfer of heat energy between two fluids. For the heat transfer to occur two fluids must be at different temperatures and they must come thermal contact. Heat exchange involve convection in each fluid and conduction through the separating wall. Heat can flow only from hotter to cooler fluids, as per the second law of thermodynamics.

Heat exchangers can be classified into four types, according to

1. Nature of heat exchange process:

• Direct contact heat exchanger: this is done by complete physical mixing of heat and mass transfer. Examples are water cooling towers and jet condensers in steam power plants.

• Regenerator: here hot and cold fluids flows alternately when hot fluid passes, the heat is transferred to the solid matrix and then stopped the flow of hot fluid, next cold fluid is passed on the matrix which takes heat from solid matrix. Examples are Open hearth and blast furnaces.

• Recuperator: the cold fluid flows simultaneously on either side of a separating wall. Examples are super heaters, condensers, economizers and air pre-heaters in steam power plants, automobile radiators.

2. Relative direction of motion of fluids:

According to flow of fluids, the Heat Exchangers are classified into three categories:

2.1 Parallel flow heat exchangers:

In parallel flow heat exchangers, both the tube side fluid and the shell side fluid flow in same direction. In this case, the two fluids enter the heat exchanger from the same end with a large temperature difference.

2.2 Counter flow heat exchangers:

In counter flow heat exchangers, the two fluids flow in opposite directions. Each of the fluids enter the heat exchanger from opposite ends. Because the cooler fluid exists the counter flow heat exchanger at the end where the hot fluid enters the heat exchanger, the cooler fluid will approach the inlet temperature of the hot fluid.

2.3 Cross flow heat exchangers:

In cross flow heat exchangers, one fluid flows through tubes and second fluid passes around the tubes perpendicularly.

3. Mechanical Design of Heat Exchanger Surface:

1. Concentric tubes
2. Shell and tube
3. Multiple shell and tube passes

4. Physical state of heat exchanging:

1. Condenser
2. Evaporator

Gear Terminology

The following are the important dimensions and geometries concerned with toothed gear:

Pitch Circle :

Pitch circle is the apparent circle that two gears can be taken like smooth cylinders rolling without friction.

Addendum Circle :

Addendum circle is the outer most profile circle of a gear. Addendum is the radial distance between the pitch circle and the addendum circle.

Dedendum Circle :

Dedendum circle is the inner most profile circle. Dedendum is the radial distance between the pitch circle and the dedendum circle.

Clearance :

Clearance is the radial distance from top of the tooth to the bottom of the tooth space in the mating gear.

Backlash :

Backlash is the tangential space between teeth of mating gears at pitch circles.

Full Depth :

Full depth is sum of the addendum and the dedendum.

Face Width :

Face width is length of tooth parallel to axes.

Diametral Pitch :

Diametral pitch (p) is the number of teeth per unit volume.

p =  (Number of Teeth) / (Diameter of Pitch circle)

Module :

Module (m) is the inverse of diametral pitch.

m = 1/p

Circular Pitch :

Circular pitch is the space in pitch circle used by each teeth.

Gear Ratio :

Gear ratio is numbers of teeth of larger gear to smaller gear.

Pressure Line :

Pressure line is the common normal at the point of contact of mating gears along which the driving tooth exerts force on the driven tooth.

Pressure Angle :

Pressure angle is the angle between the pressure line and common tangent to pitch circles. It is also called angle of obliquity. high pressure angle requires wider base and stronger teeth.

Pitch Angle :

Pitch angle is the angle captured by a tooth.

Pitch angle = 360/T

Contact Ratio :

Contact ratio is angle of angle of action and pitch angle.

Path of Approach :

Path of approach is the distance along the pressure line traveled by the contact point from the point of engagement to the pitch point.

Path of Recess :

Path of recess is the distance traveled along the pressure line by the contact point from the pitch point to the path of disengagement.

Path of Contact :

Patch of contact is the sum of path of approach and path of recess.

Arc of Approach :

Arc of approach is the distance traveled by a point on either pitch circle of the two wheels from the point of engagement to the pitch.

Arc of Recess :

Arc of recess is the distance traveled by a point on either pitch circle of the two wheels from the point to the point of disengagement.

Arc of Contact :

Arc of contact is the distance traveled by a point on either pitch circle of the two wheels during the period of contact of a pair of teeth.

Angle of Action :

Angle of action is the angle turned by a gear during arc of contact.

Classification of Gears

Gears can be classified according to relative positions of their shaft axes into three types. They are:

1. Gears for Parallel shafts

2. Gears for Intersecting Shafts

3. Gears for Skew Shafts

1. Gears for Parallel Shafts:

The motion between parallel shafts is same as to the rolling of two cylinders. Gears under this category are the following:

1.1 Spur Gears:

Straight Spur gears are the simplest form of gears having teeth parallel to the gear axis. The contact of two teeth takes place over the entire width along a line parallel to the axes of rotation. As gear rotate , the line of contact goes on shifting parallel to the shaft.

1.2 Helical Gears:

In helical gear teeth are part of helix instead of straight across the gear parallel to the axis. The mating gears will have same helix angle but in opposite direction for proper mating. As the gear rotates, the contact shifts along the line of contact in in volute helicoid across the teeth.

1.3 Herringbone Gears:

Herringbone gears are also known as Double Helical Gears. Herringbone gears are made of two helical gears with opposite helix angles, which can be up to 45 degrees.

1.4 Rack and Pinion:

In these gears the spur rack can be considered to be spur gear of infinite pitch radius with its axis of rotation placed at infinity parallel to that of pinion. The pinion rotates while the rack translates.

2. Gears for Intersecting Shafts:

The motion between two intersecting shafts is equivalent to the rolling of two cones. The gears used for intersecting shafts are called bevel gears. Gears under this category are following: 

2.1 Straight Bevel Gears:

Straight bevel gears are provided with straight teeth, radial to the point of intersection of the shaft axes and vary in cross section through the length inside generator of the cone. Straight Bevel Gears can be seen as modified version of straight spur gears in which teeth are made in conical direction instead of parallel to axis.

2.2 Spiral Bevel Gears:

Bevel gears are made with their teeth are inclined at an angle to face of the bevel. Spiral gears are also known as helical bevels.

3. Gears for Skew Shafts:

The following gears are used to join two non-parallel and non-intersecting shafts.

3.1 Hypoid Gears:

The Hypoid Gears are made of the frusta of hyperboloids of revolution. Two matching hypoid gears are made by revolving the same line of contact, these gears are not interchangeable.

3.2 Worm Gears:

The Worm Gears are used to connect skewed shafts, but not necessarily at right angles. Teeth on worm gear are cut continuously like the threads on a screw. The gear meshing with the worm gear is known as worm wheel and combination is known as worm and worm wheel.