6.1 Continuous Variable Transmissions | unit 6 stepless regulation theoretical treatment only gyroscope

TOM2

Unit -6

Step–Less-Regulation (Theoretical Treatment only) & Gyroscope

6.1 Continuous Variable Transmissions

Continuous variable transmission

The function of the transmission is to couple the engine shaft with the drive wheels, all this while making effective use of the engine torque and driving conditions.

The transmission uses a range of gears, from low to high.

Continuously variable transmission (CVT) don't have a gearbox with a set number of gears.

It rather has a set of pulley system that allows an infinite variability between highest and lowest gear shifts.

The CVT transmission allows and almost infinite number of engine speed to vehicle speed ratios.

The CVT system can be following three types:

Pulley based CVT system.

Toroidal CVT

Hydrostatic CVT

Principle of Pulley based CVT system

A pulley based CVT system is very simple in design and working.

It has only three basic components:

High power metal v belt

A variable input driving cone Pulley.

An output driven cone pulley.

Refer figure for the construction and arrangement of CVT system.

The driving pulley is connected to the crankshaft of engine, while the driven Pulley is connected to the output shaft or drive shaft of the transmission system.

The cone Pulley are the heart of the CVT system.

Each conical Pulley is made of two 20-degree cones facing each other.

The v-belt rides in the groove between the two cones.

The V belts get their name from the fact that the belts bear a V shaped cross section, which increases the frictional grip of the belt.

When the two cones of the pulleys are far apart, the belt rides lower in the groove, and the radius of the belt loop going around the pulley get smaller.

When the cones are close together, the belt rides higher in the groove, and the radius of the belt loop going around the pulley gets larger

When one Pulley increases its radius, the other decreases its radius to keep the belt tight.

As the pulley change their radii relative to one another, they create an infinite number of gear ratios from low to high and everything in between.

When the pitch radius is small on the driving Pulley and larger on the driven Pulley, then the rotational speed of the driven Pulley decreases resulting in a lower gear.

When the pitch radius is large on the driving Pulley and small on the driven Pulley, then the rotational speed of the driven Pulley increases, resulting in higher gear.

Thus, in theory, a CVT has an infinite number of gears ratios.

CVTs may use hydraulic pressure, centrifugal force or spring tension to create the force necessary to adjust the pulley halves.

CVT also have various microprocessors and sensors which continuously monitor the performance of speed of vehicle and engine, load on engine, torque requirements, etc.

Advantages of CVT system:

Reduced number of assembly/components in comparison to automatic gear transmission since no torque converter is required.

The CVT system provides smooth transition in gear ratios and hereby results in stepless acceleration from stop all the way up by cruising speed.

The CVT transmission provides less power loss than a typical automatic transmission,, resulting in better efficiency and acceleration.

Infinite gear ratios permit operation of engine at an economical operating.

Highly sophisticated gear changes: continuously variable change of gear ratios without interruption of tractive force.

Reduced engine noise: low engine speed during constant speed driving.

Enjoyable driving: a variety of electronically controlled driving programs are available.

Minimized fuel consumption.

Limitations of CVT system:

Loosening of belt tension/low life of belt limits the wide use of CVT system.

The transmission efficiency is lower than geared system due to slippage.

They have limited torque transmitting capacity in comparison of gear transmission for the same size and weight of the system.

Applications of CVT system:

A CVT system is more suitable for smaller displacement engines, especially ones with narrow power bands.

In India, CVT system is popular on gearless scooters such as Honda Activa, Honda Aviator and Suzuki access.

the CVT system is also provided as an alternative transmission on popular cars such as Nissan murano, Honda City and Audi cars

6.2 Geometry, Velocity and torque analysis of Faceplate variators

Faceplate variators are used to transmit the motion between the two perpendicular shafts.

It consists of rotating disc and moving roller refer figure.

Generally, the disc is made of steel or cast iron whereas the roller is made of cast iron and covered with a leather or plastic band.

Let, the driving shaft rotate with speed

and driving shaft with speed

in figure a

By changing the position of the roller (axially moving the roller) , the contact between roller and the disc can be changed. Hence, the transmission ratio can be changed.

The instantaneous transmission ratio for faceplate variator is given by,

Where i=instantaneous transmission ratio

speed of driving shaft

speed of driven shaft

r= contact radius of roller

R=contact radius of disc

In the above equation 1 slip between a roller and a disc is neglected. If slipped is considered, then the transmission ratio becomes

Where =Slip factor=0.97 to 0.98

Specifications

Maximum speed variation range

Maximum amount of power transmitted=4kW

Maximum amount of speed=600m/min

Efficiency=80 to 85%

The faceplate variator with to friction disc is shown in figure b. For this type of faceplate variator, the speed regulation range is wider.

In this case, by changing the position of the roller the contact between roller and both the disc can be changed. Hence the transmission ratio can be changed.

The rotating discs and roller are made to remain in contact with the help of preloaded spring.

The instantaneous transmission ratio for this arrangement is given by,

Where. r=contact radius of roller

contact radii of discs

In the above equation 3 slip, between disc and roller is neglected. If slip is considered, then the transmission ratio becomes,

Specifications

Maximum speed variation range =

Maximum amount of power transmitted =4kW

Maximum amount of speed=600m/min

Efficiency=80 to 82%

Advantages of faceplate variators

It is simple in design.

It has low cost.

It provides transmission between perpendicular shafts.

Limitation of faceplate variators

In this type of variator there is large sliding or friction loss. To overcome this drawback, we can reduce area of contact. It means, reduce the area contact to a point contact.

Application of faceplate variators

Faceplate variators are used in spindle drive of cutting of machines.

Use in camshaft drive of multiple spindle automats.

Also used in automobiles.

6.3 Conical variators

The speed variation range of conical variators and faceplate variators is same but conical variators have better velocity distribution over the area of contact.

The cone is made of steel or cast iron and friction disc is of tekstolite.

The cone variators can be employed in the following four arrangements:

Cone variator with non-parallel shafts.

Cone variator with parallel shafts.

Cone variator with planetary mechanism.

Cone variator with swivelling disc

Cone variator with non-parallel shafts

Figure shows the cone variator for transmission between non parallel shafts.

In this case, the cone is mounted on driving shaft and kept in contact with the friction disc mounted on driven shaft with the help of spring.

The cone shaft is directly connected to prime mover shaft and the speed is regulated by sliding the cone as shown in figure.

It is important to note that, the diameter of friction disc is constant (R=constant) and the transmission ratio depends on the radius of friction cone at the point of contact.

The transmission ratio is given by

Where radius of friction cone at the point of contact

R= radius of friction disc

Specifications

Maximum speed variation range=

Maximum amount of power transmitted=2.5kW

Maximum amount of speed=900m/min

Efficiency =90%

Cone variator with parallel shafts

Figure shows the cone variator for transmission between parallel shafts.

The operation of this type of variator is similar to cone variator with non-parallel shafts.

The main difference is that the speed of regulation is obtained by moving the cone inclined direction as shown in figure.

Specifications

Maximum speed variation range=

Maximum amount of power transmitted=2.5kW

Maximum amount of speed=900m/min

Efficiency=90%

Cone variator with planetary mechanism

Figure shows the cone variator with planetary mechanism.

It consists of gear A, B and the friction disc which rotates about the axis of output shaft.

This produces a self-tightening action which increases with increase in torque.

In this type of cone variator, the speed regulation can be obtained by the axial movement of cone (mounted on the motor shaft) or by the axial movement of the friction disc.

It is important note that, in this case the transmitted power is higher than the previous arrangements because of self-tightening effect.

This ensures contact between the friction disc and the cone for large torques.

Specifications

Maximum speed variation range=

Maximum amount of power transmitted=7.5kW

Maximum amount of speed=900m/min

Efficiency=85% to 90%

Cone variator with swivelling disc

Figure shows the cone variator with a swivelling spherical friction disc.

The friction disc is mounted on the driving shaft which is connected to prime mover.

Similarly, the cone is mounted on driven shaft and kept in contact with the disc by using a spring.

The speed is regulated by swivelling the friction disc about hinge hence varying the radii of cone and disc at point of contact.

Constant contact force is developed due to the weight of motor, disc and spring.

Specifications

Maximum speed variation range=

Maximum amount of power transmitted=3kW

Maximum amount of speed=900m/min

Efficiency=85%

Advantages of conical variator

Cone variator is compact in design.

In case of cone variator, friction losses are less due to point contact.

Cone variator have better velocity distribution over the area of contact.

Limitations of conical variator

Power transmission capacity is limited to 7.5kW

Cone variator is not suitable in applications where overload or running blockage occur.

Applications of conical variator

Cone variators are used in the spindle drive of small and medium-sized vertical and radial drilling machines.

Also used in automobiles.

6.4 Spheroidal and cone variators

In spheroidal and cone variator, to regulate the speed intermediate member is used.

In this type of variator, a disc of conical or spherical shape is mounted on the driving shaft and a similar disc is employed on the driven shaft.

The spheroidal and cone variators can be employed in the following two arrangements

Spheroidal variator with swivelling discs.

Cone variators with spheres supported on shafts.

Spheroidal variator with swivelling discs

This type of arrangement of spheroidal and cone variator is also called as Svetozarov variator.

Figure shows spheroidal variator with swivelling discs.

It consists of two spheroids out of which one is connected to the driving shaft and another is connected to the driven shaft.

The spheroid are made of hardened steel and the discs are of tekstolite.

In this arrangement, speed is regulated by swivelling the intermediate member about their axes.

In figure a, the contact radius of driving shaft is less than the contact radius of driven shaft. Hence, speed of driving shaft is greater than the speed of driven shaft.

In figure b both the contact radii are equal hence speed of both the shafts is also equal.

In figure c

hence

The instantaneous transmission ratio is given by,

Specifications

Maximum speed variation range

Maximum amount of power transmitted= 25kW

Maximum amount of speed=1200m/min

Efficiency=85%

Cone variators with spheres supported on shafts

Figure shows the cone variator with the sphere supported on shafts.

This type of arrangement is also called as Wulfel Kopp tourator.

It consists of two cones out of which one is mounted on the driving shaft and another is mounted on the driven shaft. Also consists of two spheres, each mounted on separate shafts.

The speed is regulated by varying the angular positions of the shafts.

This leads to a change in the effective driving radii of the spheres which act as intermediate members.

In figure a the contact radius

is less than the contact radius

hence speed of driving shaft

is greater than the speed of driven shaft

In figure b both the contact radii are equal hence speed of both the shafts is also equal

In figure c

hence

The instantaneous transmission ratio is given by

Specifications

Maximum speed variation range

Maximum amount of power transmitted=25kW

Maximum amount of speed=1200m/min

Efficiency=85%

Advantages of spheroidal and cone variators

In this variator, power transmission is maximum.

Friction losses are minimum because of point contact.

Limitations of spheroidal and cone variators

It is complicated in design.

Spheroid must be hardened to improve wear resistance.

6.5 Variators with axially displaceable cones

Figure shows the working principle of variators with axially displaceable cones.

It consists of two pairs of conical pulleys out of which one pair is mounted on the driving shaft and another pair is mounted on the driven shaft.

For sliding of these conical police, they are mounted on splined shaft.

The speed is regulated with the help of intermediate member which can be a belt, chain or a steel ring.

In figure a a pair of pulleys mounted on the driven shaft is moved closer, automatically a pair of pulleys mounted on the driving shaft moves apart.

Hence, contact radius of driving shaft is less than the contact radius of driven shaft. Therefore, speed of driving shaft is greater than the speed of driven shaft.

In figure b both the pairs of pulleys are at the same position. Hence, both the contact radii are same due to which speed of both the shafts are also same.

In figure c a pair of pulleys mounted on the driving shaft is moved closer hence a pair of pulleys mounted on the driven shaft moves apart.

Therefore

hence

The instantaneous transmission ratio is given by

This variator provides transmission ratio less than 1 ( for Speed reduction) or greater than 1 (for Speed increment.)

The specifications of this variator depends on the type of intermediate member. The commonly used intermediate members for this variator are as follows:

V belt

Nylon belt

Steel ring

Chain

V belt:-

V-belt is commonly used intermediate member which can be narrow or wide.

A narrow type of v belt is used in spindle drives of small lathe and drilling machines.

Similarly, a wide type of v belt is used in medium sized lathe and drilling machines. Also used in camshaft drive of single spindle automats.

Specifications

Maximum speed variation range

Maximum amount of power transmitted=2.5 to 15kW

Maximum amount of speed=900 to 1080m/min

Efficiency= 92%

Nylon belt

These belts are elastic hence they can be easily extended.

Nowadays a modified type of nylon belt called as cogged belt is used.

It provides positive drive between driving and driven shafts (transverse motion without slipping due to elasticity of belt).

In this case the cone pulleys are made of aluminium.

Steel ring

In this case, ring and cone pulleys are made of hardened ball bearing Steel.

It is used in spindle drive of small and medium-sized drilling machines, grinding machines, milling machines, lathe machines etc.

This type of variator is expensive because of hardened ring and conical pulleys. Also they must be manufactured accurately for proper assembly.

Specifications

Maximum speed variation range

Maximum amount of power transmitted=10kW

Maximum amount of speed=900m/min

Efficiency=85% to 90%

Chain

It can be used to provide friction type drive or to provide positive type drive.

If the chain is used as a friction type drive, then the variator is similar to that of a variator with V-belt.

The chain has rollers on the contacting surface for reducing the friction. This type of variator operates in an oil medium.

In this case the cone pulleys are made of low carbon case hardened Steel.

Specifications

Maximum speed variation range

Maximum amount of power transmitted =15kW

Maximum amount of speed=600m/min

Efficiency=90%

6.6 PIV drives

For variators with axially displaceable cons chain can be used as an intermediate member.

The chain can be used to provide friction type drive or to provide positive type drive.

If chain is used to provide positive type drive in variators with axially displaceable cons, then it is called as positive infinitely variable (PIV) drive.

Figure shows the cone pulleys and chain used in a variator with positive drive between driving and driven shafts.

It consists of a chain made of high strength alloy steel and cone police are made of low carbon case hardened Steel.

The chain consists of crosswise movable thin laminas which slide in flutes cut on the cone pulleys.

The cone Pulley with fluids act as a bevel gear.

In this drive, control screw is used to maintain the contact between axially movable cone pulleys and crosswise movable chain.

For regulation of speed, chain moves crosswise on the cone pulleys due to which contact radius between them changes.

This type of variator usually operates in an oil medium.

Applications:

PIV drive is commonly used in spindle drives of lathe machines, turrets, drilling machines, mining machines, etc.

Also used in feed drives of lathe machines and drilling machines.

6.7 Gyroscopic forces and Couples

Precessional angular motion

The angular acceleration is the rate of change of angular velocity with respect to time.

It is a vector quantity and may be represented by drawing a vector diagram with the help of right hand screw rule.

Consider a disc, as shown in figure a, revolving or spinning about the axis OX (known as axis of Spin) in anticlockwise when seen from the front, with an angular velocity

in a plane at right angles to the paper.

After a short interval of time

. Let the disc be spinning about the new axis of Spin OX’ (at an angle

) With an angular velocity (

Using the right hand screw rule, initial angular velocity of disc is represented by vector ox; and the final angular velocity of the disc is represented by vector ox' as shown in figure b

The vector xx' represents the change of angular velocity and time

i.e. the angular acceleration of the disc.

This may be resolved into two components, one parallel to ox and the other perpendicular to ox.

Component of angular acceleration in the direction of ox.

Since is very small, therefore substituting we have

In the limit, when

Component of angular acceleration in the direction perpendicular to ox,

Since is very small, therefore substituting, , we have

In the limit when

Total angular acceleration of the disc

Where

Is the angular velocity of the axis of spin about a certain Axis, which is perpendicular to the plane in which the axis of Spin is going to rotate.

This angular velocity of the axis of spin is known as angular velocity of precession and is denoted by

The axis, about which the axis of spin is to turn is known as axis of precession.

The angular motion of the axis of spin about the axis of precession is known as precessional angular motion.

If the angular velocity of the disc remains constant at all positions of the axis of Spain, then

is zero and thus

Is zero.

If the angular velocity of the disc changes the direction, but remains constant in magnitude, then angular acceleration of the disc is given by

The angular acceleration

is known as gyroscopic acceleration.

Gyroscopic couple

Consider a disc spinning with an angular velocity

rad/s about the axis of Spin OX ,in anticlockwise direction when seen from the front, as shown in figure a

Since the plane in which the disc is rotating is parallel to the plane YOZ. Therefore it is called plane of spinning.

The plane XOZ is a horizontal plane and the axis off spin rotates in a plane parallel to the horizontal plane about an axis OY. In other words, the axis of spin is said to be rotating or processing about an axis OY (which is perpendicular to both the axes OX and OZ) at an angular velocity

This horizontal plane XOZ is called plane of precession and OY is the axis of precession.

Let. I=mass moment of inertia of the disc about OX, and

= Angular velocity of the disc.

Angular momentum of the disc

Since the angular momentum is the vector quantity, therefore it may be represented by the vector

. as shown in figure b the axis of spin OX is also rotating anticlockwise when seen from the top about axis OY.

Let the axis OX is turned in the plane XOZ through a small angle

. Radians to the position OX' in time

seconds.

Assuming the angular velocity

to be constant, the angular momentum will now be represented by vector ox'.

Change in angular momentum

And rate of change of angular momentum

Since the rate of change of angular momentum will result by the application of a couple to the disc, therefore the couple applied to the disc causing precession.

Where = angular velocity of precision of the axis of spin or the speed of rotation of the axis of the spin about the axis of precision OY.

In SI units, the units of C in N-m when I is in

It may be noted that

The couple

in the direction of the vector xx' is the active gyroscopic couple, which has to be applied over the disc when the axis of spin is made to rotate with angular velocity

about the axis of precision.

When the axis of spin itself moves with angular velocity

. The disc is subjected to reactive couple whose magnitude is same but opposite in direction to that of active couple. This reactive couple to which the disc is subjected when the axis of spin rotates about the axis of precision is known as reactive gyroscopic couple. The axis of the reactive gyroscopic couple is represented by OZ' in figure a

The gyroscopic couple is usually applied through the bearings which support the shaft. The bearings will resist equal and opposite couple.

The gyroscopic principle is used in an instrument or to a known as gyroscope.

The gyroscopes are installed in ships in order to minimize the rolling and pitching effects of waves. They are also used in aeroplanes, monorail cars, gyrocompasses.

6.8 Gyroscopic stabilisation for ship and Aeroplane

The top and front view of an aeroplane are shown in figure a.

Let engine or propeller rotates in the clockwise direction when seen from the rear or tail end and the aeroplane takes a turn to the left.

Let. = Angular velocity of the engine in rad/s

m= mass of the engine and the propeller in kg.

k= its radius of gyration in metres.

I=mass momentum of inertia of the engine and the propeller in kg-m^2

v= linear velocity of the aeroplane in m/s

R= radius of curvature in metres, and

= Angular velocity of precision=v/R rad/s

Gyroscopic couple acting on the aeroplane,

Before taking the left turn, the angular momentum vector is represented by ox.

When it takes left turn, the active gyroscopic couple will change the direction of angular momentum vector from ox to ox' as shown in figure a .

The vector xx' in the limit, represents the change of angular momentum or the active gyroscopic couple and is perpendicular to ox.

Thus the plane of active gyroscopic couple XOY will be perpendicular to xx', the direction of active gyroscopic couple is clockwise as shown in the front view of figure a.

In other words, from left hand turning, the active gyroscopic couple on the aeroplane in the axis OZ will be clockwise as shown in figure (b).

The reactive gyroscopic couple will act in the opposite direction and the effect of this couple is, therefore, to raise the nose and dip the tail of the aeroplane.

When the aeroplane takes a right turn under similar conditions as discussed above, the effect of the reactive gyroscopic couple will be to dip the nose and raise the tail of the aeroplane.

When the engine or propeller rotates in anticlockwise direction when viewed from the rare or tail end the aeroplane takes a left turn, then the effect of reactive gyroscopic couple will be to dip the nose and raise the tail of the aeroplane.

When the aeroplane takes a right turn under similar conditions as mentioned in above point, the effect of reactive gyroscopic couple will be raise the nose and dip the tail of the aeroplane.

When the engine of propeller rotates in clockwise direction when viewed from the front and the aeroplane takes a left turn, then the effect of reactive gyroscopic couple will be to raise the tail and dip the nose of the aeroplane.

When the aeroplane takes a right turn under similar conditions as mentioned in above point, the effect of reactive gyroscopic couple will be to raise the nose and dip the tale of the aeroplane.

Terms used in a naval ship

The top and front views of a naval ship are shown in figure.

The fore end of the ship is called bow and the rare end is known as stern or aft.

The left hand and right hand sides of the ship, when viewed from the stern are called port and star board respectively.

Effect of gyroscopic couple on a naval ship during steering

Steering is the turning of a complete ship in a curved towards left or right, while it moves forward.

Consider the ship taking a left turn, and rotor rotates in the clockwise direction when viewed from the stern, as shown in figure.

The effect of gyroscopic couple on a naval ship during steering taking left or right on May be obtained in the similar way as for an aeroplane.

When the rotor of the ship rotates in the clockwise direction when viewed from the stern, it will have its angular momentum vector in the direction ox as shown in figure a.

As the ship steers to the left, the active gyroscopic couple will change the angular momentum vector from ox to ox'.

The vector xx' now represents the active gyroscopic couple and is perpendicular to ox .

Thus the plane of active gyroscopic couple is perpendicular to xx' and its direction in the axis OZ for left hand turn in clockwise as shown in figure.

The reactive gyroscopic couple of the same magnitude will act in the opposite direction.

The effect of this reactive gyroscopic couple is to raise the bow and lower the stern.

When the ship steers to the right under similar conditions as discussed above, the effect of the reactive gyroscopic couple as shown in figure b, will be to raise the Stern and lower the bow.

When the rotor rotates in the anticlockwise direction, when viewed from the stern and the ship is steering to the left, then the effect of reactive gyroscopic couple will be to lower the bow and raise the Stern.

When the ship is steering to the right under similar conditions as discussed in above point, then the effect reactive gyroscopic couple will be to raise the bow and lower the stern.

When the rotor rotates in the clockwise direction when viewed from the bow or fore end and the ship is steering to the left, then the effect of reactive gyroscopic couple will be to raise the Stern and lower the bow.

When the ship is steering to the right under similar conditions as discussed in above point, then the effect of reactive gyroscopic couple will be to raise the bow and lower the stern.

The effect of the reactive gyroscopic couple on a boat propelled by a turbine taking left or right turn is similar as discussed above.

Effect of gyroscopic couple on a naval ship during pitching

Pitching is the movement of a complete ship up and down in a vertical plane about transverse Axis, as shown in figure a.

In this case, the transverse axis is the axis of precession.

The pitching of the ship is assume to take place with simple harmonic motion i.e. the motion of the axis of spin about transverse axis is simple harmonic

Angular displacement of the axis for Spin from mean position after time t seconds,

= Amplitude of Swing maximum angle turned from the mean position in radians, and

= Angular velocity of S.H.M.

Angular velocity of precision

The angular velocity of precession will be maximum if =1

Maximum angular velocity of precession,

Let I=Moment of inertia of the rotor in and

= Angular velocity of the rotor in rad/sec

Maximum gyroscopic couple,

When the pitching is upward the effect the reactive gyroscopic couple, as shown in Fig (b) will try to move the ship to word star board.

On the other hand, if the pitching is downward, the effect of the reactive gyroscopic couple, as shown in figure c is to turn the ship towards port side.

Effect of gyroscopic couple on a naval ship during rolling

For the effect of gyroscopic couple to occur, the axis of precession should always be perpendicular to the axis of spin.

If, however, the axis of precession becomes parallel to the axis of Spin, there will be no effect of gyroscopic couple acting on the body of the ship.

In case of rolling of a ship, the axis of precession is always parallel to the axis of spin for all positions.

Hence, there is no effect of the gyroscopic couple acting on the body of a ship.

6.9 Stability of four wheel vehicle moving on curved path

Consider the four wheels A, B, C and D an automobile locomotive taking a turn towards left as shown in figure.

The wheels A and C are inner wheels, whereas B and D are outer wheels.

The centre of gravity of the vehicle lies vertically above the road surface.

Let m=mass of the vehicle in kg.

W= weight of the vehicle in Newton = m.g

= radius of the wheels in metres.

R= radius of curvature in metres

h= distance of centre of gravity, vertically above the road surface in metres,

x= width of track in metres,

= mass moment of inertia of one of the wheels in kg-m^3

= Angular velocity of the wheels for velocity of spin in rad/s.

= mass moment of inertia of the rotating parts of the engine

= Angular velocity of the rotating parts of the the engine in rad/s.

G= gear ratio=

v= linear velocity of the vehicle in m/s =

A little consideration will show, that the weight of the vehicle will be equally distributed over the four wheels which will act downwards. The reaction between each wheel and the road surface of the same magnitude will act upwards.

Therefore

Road reaction over each wheel =W/4=m.g/4 Newton’s

Let us now consider the effect of the gyroscopic couple and centrifugal couple on the vehicle.

Effect of the gyroscopic couple

Since the vehicle takes a turn towards left due to the precession and other rotating parts, therefore a gyroscopic couple will act.

We know that velocity of precision,

Gyroscopic couple due to four wheels

And gyroscopic couple due to rotating parts of the engine,

Net gyroscopic couple,

The positive sign is used when the wheels and rotating parts of the engine rotate in the same direction. If the rotating parts of the engine revolves in opposite direction, then negative sign is used.

Due to the gyroscopic couple, vertical reaction on the road surface will be produced.

The reaction will be vertically upwards on the outer wheels and vertically downwards on the inner wheels.

Let the magnitude of this reaction at the two outer or inner wheels be P Newton’s. Then

P × x = C. Or. P=C/x

Vertical reaction at each of the outer or inner wheels,

P/2 =C/2x

This gyroscopic couple is balanced by vertical reactions, which are vertically upwards on the outer wheels & vertically downwards on the inner wheels.

2. Effect of the centrifugal couple

Since the vehicle moves along a curved path, therefore centrifugal force will act outwardly at the centre of gravity of the vehicle.

The effect of the centrifugal force is also to overturn the vehicle.

We know that centrifugal force,

The couple tending to overturn the vehicle or overturning couple,

This overturning couple is balanced by vertical reactions, which are vertically upwards on the outer wheels & vertically downwards on the inner wheels.

Vertical reaction at each of the outer or inner wheels,

Total vertical reaction at each of the outer wheel,

And total vertical reaction at each of the inner wheel,

Stability of a two wheeler vehicle taking a turn

Consider a two wheeler taking a right turn as shown in figure

Let m=mass of the vehicle and its rider in kg.

W=weight of the vehicle and its rider in Newton’s =m.g

h= height of the centre of gravity of the vehicle and rider.

=radius of the wheels,

R= radius of track or curvature.

=mass moment of inertia of each wheel,

=mass moment of inertia of the rotating parts of the engine,

= Angular velocity of the wheels,

= Angular velocity of the engine,

G= gear ratio =

v= linear velocity of the vehicle=

= Angle of heel. It is inclination of the vehicle to the vertical for equilibrium.

Let us now consider the effect of the gyroscopic couple and centrifugal couple on the vehicle, as discussed below.

Effect of gyroscopic couple

We know that

and velocity of precession

A little consideration will show that when the wheels of move over the curved path, the vehicle is always inclined at an angle

with the vertical plane as shown in figure.

This angle is known as angle of heel. In other words, the axis of Spin is incline to the horizontal at an angle

as shown in figure.

Thus the angular momentum vector

Due to spin is represented by OA inclined to OX at an angle

But the precession axis is vertical.

Therefore the spin vector is resolved along OX.

Gyroscopic couple,

2. Effect of centrifugal couple

We know that centrifugal force,

This force acts horizontally through the centre of gravity along the outward direction,

Centrifugal couple,

Since the centrifugal couple has a tendency to overturn the vehicle, therefore total overturning couple,

= Gyroscopic couple +centrifugal couple

We know that balancing couple = m.g.h.sin

The balancing couple acts in clockwise direction when seen from the front of the vehicle.

Therefore for stability, the overturning couple must be equal to the balancing couple, i.e.

From this expression, the value of the angle of heel may be determined, so that the vehicle does not skid.

Numericals

Find the angle of inclination with respect to the vertical of a two wheeler negotiating a turn. Given combined mass of the vehicle with its rider 250kg: moment of inertia of the engine flywheel 0.3kg

. Moment of inertia of each road wheel

speed of engine flywheel 5 times that of road wheels and in the same direction; height of centre of gravity of rider with vehicle 0.6m. two wheeler speed 90 km/h; wheel radius 300 mm; radius of turn 50m.

Solution Given m=250Kg , h=0.6m . v=90km/h = 25m/s, R=50m

Let =Angle of inclination with respect to the vertical of a two wheeler

We know that gyroscopic couple,

And centrifugal couple

Total overturning couple,

We know that balancing couple,

Since the overturning couple must be equal to the balancing couple for equilibrium condition therefore

2. A ship propelled by a turbine rotor which has a mass of 5 tonnes and a speed of 2100 rpm. The rotor has a radius of gyration of 0.5 metre and rotates in clockwise direction when viewed from the stern. Find the gyroscopic effects in the following conditions:

The ship sails at a speed of 30km/h and steers to the left in a curve having 60 m radius.

The sheep pitches 6 degree above and 6 degree below the horizontal position. The bow is descending with its maximum velocity. The motion due to pitching is simple harmonic and the periodic time is 20 seconds

The ship rolls and at certain instant it has an angular velocity of 0.03 rad/s clockwise when viewed from stern.

Determine also the maximum angular acceleration during pitching. Explain how the direction of motion due to gyroscopic effect is determined in each case.

Solution Given m=5 t = 5000kg N=2100r.p.m or k=0.5m

When the ship steers to the left

Given: v=30km/h =8.33m/s : R=60m

We know that angular velocity of precession

And mass moment of inertia of the rotor

Gyroscopic couple,

When the rotor in a clockwise direction when viewed from the stern and the ship steers to the left, the effect of reactive gyroscopic couple is to raise the bow and lower the stern.

2. When the ship pitches with the bow descending

Given:

We know that angular velocity of simple harmonic motion

And maximum angular velocity of precession

Maximum gyroscopic couple

Since the ship is pitching with the bow descending therefore the effect of this maximum gyroscopic couple is to turn the ship towards port side.

3.When the ship rolls

Since the ship rolls at an angular velocity of 0.03rad/s therefore angular velocity of precession when the ship rolls,

Gyroscopic couple

In case of rolling of a ship the axis of precession is always parallel to the axis of spin for all positions, therefore there is no effect of gyroscopic couple.

Maximum angular acceleration during pitching

We know that maximum angular acceleration during pitching.

3. A four wheeled motor car of mass 2000 kg has a wheelbase 2.5 m, track with 1.5 m and height of centre of gravity 500 mm above the ground level and lies at 1 metre from the front Excel. Each wheel has an effective diameter of 0.8m and a moment of inertia of 0.8 kg- the drive shaft, engine flywheel and transmission are rotating at four times the speed of road wheel, in a clockwise direction when viewed from the front, and is equivalent to a mass of 75kg having a radius of gyration of 100mm. If the car is taking a right turn of 60m radius at 60km/h. Find the load on each wheel.

Solution Given m=2000kg: b=2.5m: x=1.5m: h=500mm = 0.5m: L=1m: R=60m: v=60km/h =16.67m/s

Since the centre of gravity of car lies at 1 m from the front axle and the weight of the car lies at the centre of gravity, therefore weight on the front wheels and rear wheels will be different.

Let Weight of the front wheels, and

=Weight on the rear wheels,

Taking moment about the front wheels,

We know that weight of the car or on the four wheels,

Weight on each of the front wheels

And weight on each of the rear wheels

Since the weight of the car over the four wheels will act downwards, therefore the reaction between each wheel and the road surface of the same magnitude will act upwards as shown in Fig.

Let us now consider the effect of gyroscopic couple due to four wheels and rotating parts of the engine.

We know angular velocity of wheels

And angular velocity of precession

Gyroscopic couple due to four wheels

This gyroscopic couple tends to lift the inner wheels and to press the outer wheels. In other words, the reaction will be vertically downward on the inner wheels and vertically upward on the outer wheels as shown in fig.

Let P/2 newtons be the magnitude of this reaction at each of the inner or outer wheel.

We know that mass moment of inertia of rotating parts of the engine,

Gyroscopic couple due to rotating parts of engine

This gyroscopic couple tends to lift the front wheels and to press the outer wheels. In other words, the reaction will be vertically downwards on the front wheels and vertically upwards on the rear wheels as shown in Fig.

Let F/2 newtons be the magnitude of this reaction on each of the front and rear wheels

Now let us consider the effect of centrifugal couple acting on the car. We know that centrifugal force,

Centrifugal couple tending to overturn the car or over turning couple,

This overturning couple tends to reduce the pressure on the inner wheels and to increase on the outer wheels. In other words, the reactions are vertically downward on the inner wheels and vertically upwards on the outer wheels. Let Q/2 be the magnitude of this reaction on each of the inner and outer wheels.

From figure we see that

Load on the front wheel 1

Load on the front wheel 2

Load on the front wheel 3

Load on the rear wheel 4

4. A rear engine automobile is travelling along a track of 100 meters mean radius. Each of the four road wheels has a moment of inertia of 2.5 and an effective diameter of 0.6m. The rotating parts of the engine have a moment of inertia of . The engine axis is parallel to the rear axle and the crankshaft rotates in the same sense as the road wheels. The ratio of engine speed to back axle speed is 3:1 .The automobile has a mass of 1600kg and has its centre of gravity 0.5m above road level. The width of the track of the vehicle is 1.5m

Determine the limiting speed of the vehicle around the curve for all four wheels to maintain contact with the road surface. Assume that the road surface is not cambered and centre of gravity of automobile lies centrally with respect to the four wheels.

Solution Given R=100m: m=1600kg: h=0.5m: x=1.5m

The weight of the vehicle will be equally distributed over the four wheels which will act downwards. The reaction between the wheel and the road surface of the same magnitude will act upwards.

Road reaction over each wheel

=W/4=m.g/4=1600*9.81/4=3924N

Let v=Limiting speed of the vehicle in m/s

We know that angular velocity of the wheels

And angular velocity of precession

Gyroscopic couple due to 4 wheels

And gyroscopic couple due to rotating parts of engine

Total gyroscopic couple

Due to this gyroscopic couple the vertical reaction on the rails will be produced . The reaction will be vertically upwards on the outer wheels and vertically downwards on the inner wheels. Let the magnitude of this reaction at each of the outer or inner wheel be P/2newtons.

We know that centrifugal force

Overturning couple acting in the outward direction

This overturning couple is balanced by vertical reactions which are vertically upwards on the outer wheels and vertically downwards on the inner wheels Let the magnitude of this reaction at each of the outer or inner wheels be Q/2 newtons.

We know that total vertical reaction at each of the outer wheels

And total vertical reaction at each of the inner wheels

From equation (i) we see that there will always be contact between the outer wheels and the road surface because W/4 P/2 and Q/2 are vertically upwards, In order to have contact between the inner wheels and road surface the reactions should also be vertically upwards which is only possible if

Reference:

Theory of machines – Khurmi & Gupta

Theory of machines – Anup Goel

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