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Unit 5 Gyroscope, , T.Y.BTech (Mech), , Content, Gyroscope: Principles of gyroscopic action, Precession and gyroscopic acceleration,, gyroscopic couple, Effect of the gyroscopic couple on ships, aero planes and vehicles,, inclined rotating discs, gyroscopic stabilization., , 5.1 Introduction, ‘Gyre’ is a Greek word, meaning ‘circular motion’ and Gyration means the whirling, motion. A gyroscope is a spatial mechanism which is generally employed for the study, of precessional motion of a rotary body. Gyroscope finds applications in gyrocompass,, used in aircraft, naval ship, control system of missiles and space shuttle. The gyroscopic, effect is also felt on the automotive vehicles while negotiating a turn. A gyroscope, consists of a rotor mounted in the inner gimbal. The inner gimbal is mounted in the outer, gimbal which itself is mounted on a fixed frame as shown in Fig.5.1. When the rotor, spins about X-axis with angular velocity ω rad/s and the inner gimbal precesses (rotates), about Y-axis, the spatial mechanism is forced to turn about Z-axis other than its own, axis of rotation, and the gyroscopic effect is thus setup. The resistance to this motion is, called gyroscopic effect., , Fig 5.1 Gyroscopic mechanism, , 5.2 Principle of Gyroscope, If the axis of spinning or rotating body is given an angular motion about an axis, perpendicular to the axis of spin, an angular acceleration acts on the body about the third, perpendicular axis. The torque required to produce this acceleration is known as active, gyroscopic torque. A reactive gyroscopic torque or couple also acts similar to the, concept of centripetal and centrifugal forces on a reacting body. The effect produced by, Prof. Sachin M.Shinde,KECSP, , Page 1

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Unit 5 Gyroscope, , T.Y.BTech (Mech), , the reactive gyroscopic couple is known as gyroscopic effect. Thus aero planes, ships,, automobiles, etc., that have rotating parts in the form of wheels or rotors of engines, experiences this effect while taking turn, i.e., when the axes of spin is subjected to some, angular motion, , 5.3 Angular Velocity, The angular velocity of a rotating body is specified by, a., , the magnitude of velocity, , b., , the direction of the axis of rotor, , c., , the sense of rotation of the rotor, i.e., clockwise or counter-clockwise, , Angular velocity is represented by a vector in the following manner:, (i) Magnitude of the velocity is represented by the length of the vector., (ii) Direction of axis of the rotor is represented by drawing the vector parallel to the axis, of the rotor or normal to the plane of the angular velocity., (iii) Sense of rotation of the rotor is denoted by taking the direction of the vector in a, set rule., The general rule is that of a right-handed screw, i.e., if a screw is rotated in the, clockwise direction, it goes away from the viewer and vice-versa., , Fig 5.2, For example, Fig. 5.2 (a) shows a rotor which rotates in the clockwise direction when, viewed from the end l. Its angular motion has been shown vectorially in Fig.5.2 (b). The, vector has been taken to a scale parallel to the axis of the rotor. The sense of direction of, the vector is from a to b according to the screw rule. However, if the direction of rotation, of the rotor is reversed, it would be from b to a [Fig.5.2 (c)]., Prof. Sachin M.Shinde,KECSP, , Page 2

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Unit 5 Gyroscope, , T.Y.BTech (Mech), , 5.4 Angular Acceleration, Let a rotor spin (rotate) about the horizontal axis Ox at a speed of ω rad/s in the direction, as shown in Fig.5.3 (a). Let oa represent its angular velocity Fig. 5.3 (b)., , Fig 5.3 Angular Acceleration, Now, if the magnitude of the angular velocity changes to (ω+δω) and the direction of the, axis of spin to Ox’ (in time δt). The vector ob would represent its angular velocity in the, new position. Join ab which represents the change in the angular velocity of the rotor., The vector ab can be resolved into two components:, (i) ac representing angular velocity change in a plane normal to ac or x-axis, and, (ii) cb representing angular velocity change in a plane normal to cb or y-axis., Change of angular velocity,, 𝑎𝑐 = (𝜔 + 𝛿𝜔) 𝑐𝑜𝑠 𝛿𝜃 – 𝜔, Rate of change of angular velocity, =, Angular acceleration, 𝜔, , 𝛿𝜔 𝑐𝑜𝑠 𝛿𝜃, 𝛿, , 𝜔𝛿, , 𝐴𝑠 𝛿 → 0, 𝛿𝜃 → 0 𝑎𝑛𝑑 cos 𝛿𝜃 → 1, Angular acceleration, =, Change of angular velocity, Prof. Sachin M.Shinde,KECSP, , Page 3

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Unit 5 Gyroscope, , T.Y.BTech (Mech), 𝑐𝑑 = (𝜔 + 𝛿𝜔) sin 𝛿𝜃, , Rate of change of angular velocity = ((ω + δω) sin δθ) / δt, Angular acceleration =, 𝜔, , 𝛿𝜔 𝑠 𝑛 𝛿𝜃, 𝛿, , 𝐴𝑠 𝛿 → 0, 𝛿𝜃 → 0 𝑎𝑛𝑑 cos 𝛿𝜃 → 𝛿𝜃, Angular acceleration =, =ω, Total angular acceleration,, α=, This shows that the total angular acceleration of the rotor is the sum of, 1. dω/dt, representing change in the magnitude of the angular velocity of the rotor, 2. ω dθ/dt, Representing change in the direction of the axis of spin, the direction of cb is, from c to b in the vector diagram (being a component of ab), the acceleration acts, clockwise in the vertical plane xy. (When viewed from front along they-axis), , 5.5 Gyroscopic Torque (Couple), Let I be the moment of inertia of a rotor and 𝜔 its angular velocity about a horizontal, axis of spin Ox in the direction as shown in Fig.5.4 (a). Let this axis of spin turn through, a small angle 𝛿𝜃 in the horizontal plane (xy) to the position Ox' in time δt,, , Figure 5.4 Gyroscopic Torques (Couple), , Prof. Sachin M.Shinde,KECSP, , Page 4

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Unit 5 Gyroscope, , T.Y.BTech (Mech), , Figure 5.4 (b) shows the vector diagram. oa represents the angular velocity vector when, the axis is ox and ob when the axis is changed to ox'. Then ab represents the change in, the angular velocity due to change in direction of the axis of spin of the rotor. This, change in the angular velocity is clockwise when viewed from a towards b and is in the, vertical plane xz. This change results in angular acceleration, the sense and direction of, which are the same as that of the change in the angular velocity., Change in the angular velocity,, 𝒂𝒃 = 𝜔 × 𝛿𝜃, Angular acceleration,, α=, In the limit,, When, 𝛿 → 0, α =, Usually, dθ/dt the angular velocity of the axis of spin is called the angular velocity of, precession and is denoted by ωp, Angular acceleration, 𝛼 = 𝜔 ∙ 𝜔p, The torque required to produce this acceleration is known as the gyroscopic torque and, is a couple which must be applied to the axis of spin to cause it to rotate with angular, velocity ωp about the axis of precession Oz., Acceleration torque,, 𝑇 = 𝐼 ∙ 𝛼 = 𝐼 ∙ 𝜔 ∙ 𝜔p, For the configuration of Fig.(a), Ox is known as the axis of spin, Oz is known as the axis of precession, Oy is known as the axis of gyroscopic couple 00, yz is plane of spin, xy is plane of precession, yz is plane of gyroscopic couple, The torque obtained above is that which is required to cause the axis of spin to, precesses in the horizontal, , plane and is known as the active gyroscopic torque or the, , applied torque. A re- active gyroscopic torque or reaction torque is also applied to the, axis which tends to rotate the axis of spin in the opposite direction i.e., in the, Prof. Sachin M.Shinde,KECSP, , Page 5

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Unit 5 Gyroscope, counter-clockwise, rotating, , T.Y.BTech (Mech), , direction in the above case. Just as the centrifugal force on a, , body tends move the body tends to move the body outwards,, , while a, , centripetal acceleration (and thus centripetal force) acts on it in- wards, in the same, way, the effects of active and reactive gyroscopic torques can be understood., The effect of the gyroscopic couple on a rotating body is known as the gyroscope effect, on the body. A gyroscope is a spinning body which is free to move in other directions, under the action of external forces., , 5.6 Direction of Spin vector, Precession vector and Couple/Torque, vector with forced precession, , Fig 5.5 Direction of Spin vector, Precession vector and Couple/Torque vector with, forced precession, To determine the direction of spin, precession and torque/couple vector, right hand, screw rule or right hand rule is used. The fingers represent the rotation of the disc and, the thumb shows the direction of the spin, precession and torque vector (Fig.5.5)., The method of determining the direction of couple/torque vector is as follows, Case (i):, Consider a rotor rotating in anticlockwise direction when seen from the right (Fig.5.6, and Fig. 5.7), and to precesses the spin axis about precession axis in clockwise and, anticlockwise direction when seen from top. Then, to determine the active/reactive, gyroscopic couple vector, the following procedure is used., Prof. Sachin M.Shinde,KECSP, , Page 6

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Unit 5 Gyroscope, , T.Y.BTech (Mech), , • Turn the spin vector through 900 in the direction of precession on the XOZ plane, The turned spin vector will then correspond to the direction of active gyroscopic, Couple/torque vector, • The reactive gyroscopic couple/torque vector is taken opposite to active gyro vector, direction, , Fig 5.6 Direction of active and reactive Couple/Torque vector, , Fig 5.7 Direction of active and reactive Couple/Torque vector, Case (ii):, Consider a rotor rotating in clockwise direction when seen from the right (Fig.5.8 and, Fig. 5.9), and to precesses the spin axis about precession axis in clockwise and, , Prof. Sachin M.Shinde,KECSP, , Page 7

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Unit 5 Gyroscope, , T.Y.BTech (Mech), , anticlockwise direction when seen from top. Then, to determine the active/reactive, gyroscopic couple vector,, • Turn the spin vector through 900 in the direction of precession on the XOZ plane, The turned spin vector will then correspond to the direction of active gyroscopic, Couple/torque vector, • The reactive gyroscopic couple/torque vector is taken opposite to active gyro vector, direction, , Fig 5.8 Direction of active and reactive Couple/Torque vector, , Fig 5.9 Direction of active and reactive Couple/Torque vector, The resisting couple/ reactive couple will act in the direction opposite to that of the, gyroscopic couple. This means that, whenever the axis of spin changes its direction, a, gyroscopic couple is applied to it through the bearing which supports the spinning axis., Prof. Sachin M.Shinde,KECSP, , Page 8

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Unit 5 Gyroscope, , T.Y.BTech (Mech), , 5.7 Gyroscopic Effect on Ship, Gyroscope is used for stabilization and directional control of a ship sailing in the rough, sea. A ship, while navigating in the rough sea, may experience the following three, different types of motion:, (i) Steering—The turning of ship in a curve while moving forward, (ii) Pitching—The movement of the ship up and down from horizontal position in a, vertical plane about transverse axis, (iii)Rolling—sideway motion of the ship about longitudinal axis, For stabilization of a ship against any of the above motion, the major requirement is that, the gyroscope shall be made to precesses in such a way that reaction couple exerted by, the rotor opposes the disturbing couple which may act on the frame., Ship Terminology, (i) Bow – It is the fore end of ship, (ii) Stern – It is the rear end of ship, (iii) Starboard – It is the right hand side of the ship looking in the direction of motion, (iv) Port – It is the left hand side of the ship looking in the direction of motion, , Fig 5.10 Terms used in a naval ship., Consider a gyro-rotor mounted on the ship along longitudinal axis (X-axis) as shown in, Fig.5.10 and rotate in clockwise direction when viewed from rear end of the ship. The, Prof. Sachin M.Shinde,KECSP, , Page 9

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Unit 5 Gyroscope, , T.Y.BTech (Mech), , angular speed of the rotor is ω rad/s. The direction of angular momentum vector oa,, based on direction of effect during the three types of motion of ship is discussed., Gyroscopic effect on Steering of ship, (i) Left turn with clockwise rotor, When ship takes a left turn and the rotor rotates in clockwise direction viewed from, stern, the gyroscopic couple act on the ship is analyzed in the following way, Note that, always reactive gyroscopic couple is considered for analysis. From the above, analysis (Fig.5.11), the couple acts over the ship between stern and bow. This reaction, couple tends to raise the front end (bow) and lower the rear end (stern) of the ship., , Fig 5.11, Prof. Sachin M.Shinde,KECSP, , Page 10

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Unit 5 Gyroscope, , T.Y.BTech (Mech), , (ii) Right turn with clockwise rotor, When ship takes a right turn and the rotor rotates in clockwise direction viewed from, stern, the gyroscopic couple acts on the ship is analyzed (Fig 5.12). Again, the couple, acts in vertical plane, means between stern and bow. Now the reaction couple tends to, lower the bow of the ship and raise the stern, , Fig 5.12, (iii)Left turn with anticlockwise rotor, When ship takes a left turn and the rotor rotates in anticlockwise direction viewed from, stern, the gyroscopic couple act on the ship is analyzed in the following way (Fig.5.13)., The couple acts over the ship are between stern and bow. This reaction couple tends to, press or dip the front end (bow) and raise the rear end (stern) of the ship., Prof. Sachin M.Shinde,KECSP, , Page 11

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Unit 5 Gyroscope, , T.Y.BTech (Mech), , (iv) Right turn with anticlockwise rotor, When ship takes a right turn and the rotor rotates in anticlockwise direction viewed, from stern, the gyroscopic couple act on the ship is according to Fig 5.14. Now, the, reaction couple tends to raise the bow of the ship and dip the stern, , Fig 5.14, Fig 5.13, , Prof. Sachin M.Shinde,KECSP, , Page 12

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Unit 5 Gyroscope, , T.Y.BTech (Mech), , Gyroscopic effect on Pitching of ship, The pitching motion of a ship generally occurs due to waves which can be, approximated as sine wave. During pitching, the ship moves up and down from the, horizontal position in vertical plane (Fig.5.15), , Fig 5.15, ∴ Angular displacement of the axis of spin from mean position after time t seconds,, θ= φ sin ω1. t, Where φ = Amplitude of swing i.e. maximum angle turned from the mean position in, radians, and, ω1 = Angular velocity of S.H.M., =, , =, , rad/s, , =, , (φ sin ω1.t) = φ cos ω1.t, , Angular velocity of precession, ωp =, , The angular velocity of precession will be maximum, if cos ω1.t = 1., ∴ Maximum angular velocity of precession,, ωPmax= φ.ω1 = φ × 2π / tp ...(Substituting cos ω1.t = 1), Let I = Moment of inertia of the rotor in kg-m2, and, ω = Angular velocity of the rotor in rad/s., ∴ Minimum gyroscopic couple,, Cmax = I. ω. ωPmax, Consider a rotor mounted along the longitudinal axis and rotates in clockwise direction, when seen from the rear end of the ship. The direction of momentum for this condition, is shown by vector ox (Fig.5.16). When the ship moves up the horizontal position in, vertical plane by an angle δθ from the axis of spin, the rotor axis (X-axis) processes, Prof. Sachin M.Shinde,KECSP, , Page 13

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Unit 5 Gyroscope, , T.Y.BTech (Mech), , about Z- axis in XY-plane and direction about Y-axis and the reaction couple acts in, opposite direction, i.e. in clockwise direction, which tends to move towards right side, (Fig.5.16). However, when the ship pitches down the axis of spin, the direction of, reaction couple is reversed and the ship turns towards left side (Fig.5.17), , Fig 5.16, , Fig 5.17, , Similarly, for the anticlockwise direction of the rotor viewed from the rear end (Stern) of, the ship, the analysis may be done., Gyroscopic effect on Rolling of ship, The axis of the rotor of a ship is mounted along the longitudinal axis of ship and, therefore, there is no precession of this axis. Thus, no effect of gyroscopic couple on the, ship frame is formed when the ship rolls, , 5.8 Gyroscopic Effect on Aero plane, Aero planes are subjected to gyroscopic effect when it taking off, landing and, negotiating left or right turns in the air., Let, ω = Angular velocity of the engine rotating parts in rad/s,, m = Mass of the engine and propeller in kg,, rW = Radius of gyration in m,, I = Mass moment of inertia of engine and propeller in kg m2,, V = Linear velocity of the aeroplane in m/s,, Prof. Sachin M.Shinde,KECSP, , Page 14

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Unit 5 Gyroscope, , T.Y.BTech (Mech), , R = Radius of curvature in m,, ωp =Angular velocity of precession =v/R rad/s, Gyroscopic couple acting on the aero plane = C = I ω ωp, Let us analyze the effect of gyroscopic couple acting on the body of the aero plane for, various conditions., Case (i): propeller rotates in clockwise direction when seen from rear end and aero, plane turns towards left as shown if fig 5.18, , Fig 5.18, Prof. Sachin M.Shinde,KECSP, , Page 15

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Unit 5 Gyroscope, , T.Y.BTech (Mech), , According to the analysis, the reactive gyroscopic couple tends to dip the tail and raise, the nose of aero plan as shown below in fig 5.19, , Fig 5.19, Case (ii): propeller rotates in clockwise direction when seen from rear, end and aero plane turns towards right as shown in fig 5.20,5.21, , Fig 5.20, Prof. Sachin M.Shinde,KECSP, , Page 16

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Unit 5 Gyroscope, , T.Y.BTech (Mech), , Fig 5.21, According to the analysis, the reactive gyroscopic couple tends to raise the tail and dip, the nose of aero plane, , Fig 5.22, Case (iii): propeller rotates in anticlockwise direction when seen from, rear end and aeroplane turns towards left as shown in fig 5.22,5.23, Prof. Sachin M.Shinde,KECSP, , Page 17

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Unit 5 Gyroscope, , T.Y.BTech (Mech), , Fig 5.22, The analysis indicates, the reactive gyroscopic couple tends to raise the tail and dip the, nose of aero plane, , Fig 5.23, Prof. Sachin M.Shinde,KECSP, , Page 18

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Unit 5 Gyroscope, , T.Y.BTech (Mech), , Case (iv): propeller rotates in anticlockwise direction when seen from rear end and aero, plane turns towards right as shown in fig 5.24, 5.25, , Fig 5.24, Prof. Sachin M.Shinde,KECSP, , Page 19

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Unit 5 Gyroscope, , T.Y.BTech (Mech), , The analysis shows, the reactive gyroscopic couple tends to raise the tail and dip the, nose of aero plane, , Fig 5.25, Case (v): propeller rotates in clockwise direction when seen from rear end and aero, plane takes off or nose move upwards as shown in fig 5.26,5.27, , Fig 5.26, , Fig 5.27, The analysis show, the reactive gyroscopic couple tends to turn the nose of aero plane, toward right, , Prof. Sachin M.Shinde,KECSP, , Page 20

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Unit 5 Gyroscope, , T.Y.BTech (Mech), , Case (vi): propeller rotates in clockwise direction when seen from rear end and aero, plane is landing or nose move downwards as shown in fig 5.28, , Fig 5.28, The reactive gyroscopic couple tends to turn the nose of aero plane toward left, Case (vii): propeller rotates in anticlockwise direction when seen from, , rear end and, , aero plane takes off or nose move upwards as shown in fig 5.29, 5.30, , Fig 5.29, Prof. Sachin M.Shinde,KECSP, , Page 21

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Unit 5 Gyroscope, , T.Y.BTech (Mech), , The reactive gyroscopic couple tends to turn the nose of aero plane toward left as shown, in fig 5.30, , Fig 5.30, Case (viii): propeller rotates in anticlockwise direction when seen from rear end and, aero plane is landing or nose move downwards. The analysis show, the reactive, gyroscopic couple tends to turn the nose of aero plane toward right as shown in fig 5.31, , Fig 5.31, , 5.8 Stability of a four wheel drive Moving in a Curved Path, Consider the four wheels A, B, C and D of an automobile locomotive taking a turn, towards left as shown in Fig. 5.32. The wheels A and C are inner wheels, whereas B and, D are outer wheels. The centre of gravity (C.G.) of the vehicle lies vertically above the, road surface., Let m = Mass of the vehicle in kg,, W = Weight of the vehicle in newton’s = m.g,, rW = Radius of the wheels in metres,, Prof. Sachin M.Shinde,KECSP, , Page 22

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Unit 5 Gyroscope, , T.Y.BTech (Mech), , R = Radius of curvature in meters (R > rW),, h = Distance of centre of gravity, vertically above the road surface in meters,, x = Width of track in meters,, IW = Mass moment of inertia of one of the wheels in kg-m2,, ωW = Angular velocity of the wheels or velocity of spin in rad/s,, IE = Mass moment of inertia of the rotating parts of the engine in kg-m2,, ωE = Angular velocity of the rotating parts of the engine in rad/s,, G = Gear ratio = ωE /ωW,, v = Linear velocity of the vehicle in m/s = ωW.rW, , Fig. 5.32. Four wheel drive moving in a curved path, A little consideration will show, that the weight of the vehicle (W) 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., 1. Effect of the gyroscopic couple, Since the vehicle takes a turn towards left due to the precession and other rotating parts,, Prof. Sachin M.Shinde,KECSP, , Page 23

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Unit 5 Gyroscope, , T.Y.BTech (Mech), , Therefore a gyroscopic couple will act. We know that velocity of precession,, , ωP = v/R, ∴ Gyroscopic couple due to 4 wheels,, , CW =4 IW.ωW.ωP, and gyroscopic couple due to the rotating parts of the engine,, , CE = IE.ωE.ωP = IE.G.ωW.ωP ... (G = ωE/ωW), ∴ Net gyroscopic couple,, , C = CW ± CE = 4 IW.ωW.ωP ± IE.G.ωW.ωP, = ωW.ωP (4 IW ± G.IE), 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 revolve 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. Then, P × x = C or P = C/x, ∴ Vertical reaction at each of the outer or inner wheels,, P /2 = C/ 2x, Note: We have discussed above that when rotating parts of the engine rotate in opposite, directions, then –ve sign is used, i.e. net gyroscopic couple,, C = CW – CE, When CE > CW, then C will be –ve. Thus the reaction will be vertically downwards on, the outer wheels and vertically upwards 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 this centrifugal force is, also to overturn the vehicle., We know that centrifugal force,, , Prof. Sachin M.Shinde,KECSP, , Page 24

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Unit 5 Gyroscope, ∴, , The, , couple, , T.Y.BTech (Mech), , tending, , to, , overturn, , the, , vehicle, , or, , overturning, , couple,, , .h, 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 the two outer or inner wheels be Q. Then, , Q × x = CO or Q =, , =, , .h, , ∴ Vertical reaction at each of the outer or inner wheels,, , =, , .h, , ∴ Total vertical reaction at each of the outer wheel,, , and total vertical reaction at each of the inner wheel,, , A little consideration will show that when the vehicle is running at high speeds, P I may, be zero or even negative. This will cause the inner wheels to leave the ground thus, tending to overturn the automobile. In order to have the contact between the inner, wheels and the ground, the sum of P/2 and Q/2 must be less than W/4., , 5.9 Stability of a Two Wheel Vehicle Taking a Turn, Consider a two wheel vehicle (say a scooter or motor cycle) taking a right turn as shown, in Fig. 5.33 (a)., Let m = Mass of the vehicle and its rider in kg,, W = Weight of the vehicle and its rider in newton = m.g,, h = Height of the centre of gravity of the vehicle and rider,, rW = Radius of the wheels,, R = Radius of track or curvature,, IW = Mass moment of inertia of each wheel,, IE = Mass moment of inertia of the rotating parts of the engine,, ωW = Angular velocity of the wheels,, ωE = Angular velocity of the engine,, Prof. Sachin M.Shinde,KECSP, , Page 25

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Unit 5 Gyroscope, , T.Y.BTech (Mech), , G = Gear ratio = ωE / ωW,, v = Linear velocity of the vehicle = ωW × rW,, θ = 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., , Fig 5.33 Stability of a two wheel vehicle taking a turn, 1.Effect of gyroscopic couple, We know that, , and, , ∴ Total, , and velocity of precession, ωP = v /R, A little consideration will show that when the wheels move over the curved path, the, vehicle, Prof. Sachin M.Shinde,KECSP, , Page 26

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Unit 5 Gyroscope, , T.Y.BTech (Mech), , is always inclined at an angle θ with the vertical plane as shown in Fig. 5.33 (b). This, angle is known as angle of heel. In other words, the axis of spin is inclined to the, horizontal at an angle θ, as shown in Fig. 5.33 (c). Thus the angular momentum vector, Iω due to spin is represented by OA inclined to OX at an angle θ. But the precession axis, is vertical. Therefore the spin vector is resolved along OX., ∴ Gyroscopic couple,, , Notes: (a) When the engine is rotating in the same direction as that of wheels, then the, positive sign is used in the above expression and if the engine rotates in opposite, direction, then negative sign is used., (b) The gyroscopic couple will act over the vehicle outwards i.e. in the anticlockwise, direction when seen from the front of the vehicle. The tendency of this couple is to, overturn the vehicle in outward direction., 2. Effect of centrifugal couple, We know that centrifugal force,, , This force acts horizontally through the centre of gravity (C.G.) along the outward, direction., ∴ Centrifugal couple,, , Since the centrifugal couple has a tendency to overturn the vehicle, th, Total overturning couple,, CO = Gyroscopic couple + Centrifugal couple, , We know that balancing couple = m.g.h sin θ, Prof. Sachin M.Shinde,KECSP, , Page 27

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Unit 5 Gyroscope, , T.Y.BTech (Mech), , 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 (θ) may be determined, so that the, vehicle does not skid., , Prof. Sachin M.Shinde,KECSP, , Page 28