2. Gyroscopic couple = I* ω * ωp N-m
Where, I is moment of inertia of disc in kg-m2ω is angular velocity of disc in rad/sec
ωp is velocity of precession rad/sec
Theoretical torque / couple is N-m
Gyroscopic couple is N-m
From the above results it can be concluded that the theoretical and practical couples produced
Conform to the principle of gyroscope.
Increase the speed of the motor gradually in the range given.
Do not add large weight on the weight pan.
Always maintain safe distance from the apparatus.
EXPERIMENT – 06
AIM: To determine the frequency of undamped free vibration of an equivalent spring mass system.
THEORY: The vibrations the system executes under no damping condition is known as undamped vibrations. Neglecting damping is also considered as undamped situation. When no external force is acts on the body after giving an initial displacement then the body is said to be under free or natural vibrations. The frequency of the free vibrations is called free or natural frequency and denoted by fn. simple pendulum is an example of undamped free vibrations.
FIGURE: Undamped free vibration arrangement as fig 6.1
Support one end of the beam in the slot of trunion and clamp it by means of screw.
Attach the other end of the beam to the lower end of the spring.
Set the beam in the horizontal position.
Measure the distance LI of the assembly from pivot.
Allow the system to vibrate.
Measure the time for say 10 oscillations and find the periodic time and natural • frequency of vibration.
Repeat the experiment by varying L1.
(T = t/n)
(fn = 1/t)
CALCULATIONS: Periodic time T (theoretical)
where rrie equivalent mass at the spring = m (L12 / L2) K = stiffness of the spring 0.3 kg/mm
m = (W + w) / g
w = weight attached to exciter assembly W = weight exciter assembly = 4.44 kg LI distance of W from pivot = 0.25 m
L is distance of spring from pivot 0.94 m
The theoretical natural frequency is
The experimental natural frequency is
It is to conclude that the theoretical and experimental natural frequency of vibration is almost equal.
EXPERIMENT NO. 7
AIM: To study the various types of kinematic links, pairs, chains and mechanisms using models.
KINEMATIC LINK: - A mechanism is made of a number of resistant bodies out of which some may have motions relative to the others. A resistant body or a group of resistant bodies with rigid connections preventing their relative movement is known as a link. A link is also known as kinematic link or element.
Examples: - A slider-crank mechanism (figure 1.1) consists of four links: frame and guides, crank connecting rod and slider, the crank link may have crankshaft and flywheel also, forming one link having no relative motion of these.
Figure 7.1 Slider-crank mechanism
CLASSIFICATIONS OF LINKS:- Binary link.
KINEMATIC PAIR: - A kinematic pair or simply a pair is a joint of two links having relative motion between them.
CLASSIFICATIONS OF PAIRS: Kinematics pairs according to nature of contact:-
Lower pair (links have surface or area contact).
Examples- Nut turning on a screw, shaft rotating in a bearing, universal joint etc.
Higher pair (Point or line contact between the links).
Examples: - When rolling on a surface, cam and follower pair, tooth gears, ball and roller bearings etc.
Kinematics pairs according to nature of mechanical constraint:-
Closed pair (when the elements of a pair are held together mechanically). Examples: - all the lower pairs and some of the higher pair.
Unclosed pair (when two links of a pair are in contact either due to force of gravity or some spring action).
Kinematics pairs according to nature of relative motion:-
Sliding or Prismatic pair
As shown in Figure 7.4 (a), a prismatic pair allows only a relative translation between elements 1 and 2, which can be expressed by a single coordinate ‘s’, and it has one degree of freedom.
Figure 7.4 (a) Prismatic Pair
Turning or Revolute pair
A revolute pair is shown in Figure 7.4 (b). It is seen that this pair allows only one relative rotation between elements 1 and 2, which can be expressed by a single coordinate ‘θ’. Thus, a revolute pair has a single degree of freedom.
Figure 7.4 (b) Revolute Pair
Rolling or Cylindrical Pair
As shown in Figure, a cylindrical pair allows both rotation and translation parallel to the axis of rotation between elements 1 and 2. These relative movements can be expressed by two independent coordinates ‘θ’ or‘s’ because they are not related with each other. Degrees of freedom in this case are equal to two.
Figure 7.4 (c) Rolling Pair
Screw pair (Helical pair)
As shown in Figure 7.4 (d), a screw pair allows rotation as well as translation but these two movements are related to each other. Therefore, screw pair has one degree of freedom because the relative movement between 1 and 2 can be expressed by a single coordinate ‘θ’ or‘s’.
Figure 7.4 (d) Screw Pair
A ball and socket joint, as shown in Figure 7.4 (e), forms a spherical pair. Any rotation of element 2 relative to 1 can be resolved in the three components. Therefore, the complete description of motion requires three independent coordinates. Two of these coordinates ‘α’ and ‘β’ are required to specify the position of axis OA and the third coordinate ‘θ’ describes the rotation about the axis of OA. This pair has three degrees of freedom.
Figure 7.4 (e) Spherical Pair
KINEMATIC CHAIN: - A kinematic chain is an assembly of links in which the relative motions of the links is possible and the motion of each relative to the others is definite. If indefinite motions of other links, it is a non-kinematic chain.
Types of kinematics chains:-
Four bar chain or quadric cycle chain
Single slider crank chain
Double slider crank chain
MECHANISM: - A linkage is obtained if one of the links of a kinematics chain is fixed to the ground. If motion of each link, results in definite motions of the others, the linkage is known as a mechanism. If one of the links of a redundant chain is fixed, it is known as a structure. The degree of freedom of a structure is zero or less. A structure with negative degree of freedom is known as a superstructure.
OBSERVATION & CONCLUSION:-
Comparison between kinematics links, Pairs, chains & Mechanisms.
Type of Motion to be named.
EXPERIMENT NO. 8
AIM: - To study of inversions of four-bar mechanisms, Single & double slider Crank mechanisms, using models.
APPARATUS USED: - Models of four-Bar Mechanisms, Single & Double slider crank mechanisms.
FOUR BAR MECHANISM: - A four bar link mechanism or linkage is the most fundamental of the plane kinematics linkages. It is a much preferred mechanical device for the mechanization and control of motion due to its simplicity and versatility. Basically it consists of four rigid links which are connected in the form of a quadrilateral by four pin joints. A link that makes complete revolutions is the crank, the link opposite to the fixed link is the coupler and the fourth link a lever or rocker if oscillates or an another crank, if rotate.
By fixing the link:-
Shortest Link Fixed.
Link opposite to Shortest Link fixed.
Figure 8.1 Four bar mechanism
Inversions of 4-Bar Chain Double Crank mechanism
Crank-Rocker or Rocker Crank Mechanism
Double Rocker Mechanism
Examples of Inversions of 4-Bar chain
Double Crank Mechanism (Coupling Rod of Locomotive)
In this mechanism, the links AB and DC acts as cranks and are connected to the respective wheels. The link BC acting as a coupling rod and the link AD is a fixed link in order to maintain constant centre to centre distance between them. This mechanism is meant for transmitting rotary motion from one wheel to another wheel.
Figure 8.2(a) Locomotive Coupler
Beam Engine (Crank-Rocker or Rocker-Crank)
A part of the mechanism of a beam engine (also known as crank and lever mechanism) which consist of four links. In this mechanism when the crank rotates about the fixed centre A, the lever oscillates about the fixed centre D.
Figure 8.2 (b) Beam Engine
Watt’s mechanism (Double Rocker Mechanism)
It is a crossed four bar chain mechanism and was used by watt for the early steam engines to
guide the piston rod in a cylinder to have an approximate straight line motion.
Figure 8.2(c) Watt’s mechanism
INVERSIONS OF SINGLE SLIDER–CRANK CHAIN:- Different mechanisms obtained by fixing different links of a kinematics chain are known as its inversions.
A slider –crank chain has the following inversions:-
1. First inversion (i.e.; Reciprocating engine and compressor) – this inversion is obtained when link 1 is fixed and links 2 and 4 are made the crank and the slider respectively.
Figure 8.3 Mechanism used in (a) Reciprocating engine; (b) Reciprocating compressor
a) Second inversion (i.e., Whitworth quick return mechanism and Rotary engine) – fixing of link 2 of a slider – crank chain.
Figure 8.4(a) Whitworth quick return mechanism
Figure 8.4(b) Rotary engine
Third inversion (i.e., Oscillating cylinder engine and crank & slotted – lever mechanism) - By fixing link 3 of the slider crank mechanism.
Figure 8.5(a) Oscillating cylinder engine (b) Slotted – lever mechanism
Fourth inversion (Hand pump) – if link 4 of the slider crank mechanism is fixed, the fourth inversion is obtained.
Figure 8.6 Hand pump
DOUBLE-SLIDER CRANK-CHAIN: A four-bar chain having two turning and two sliding pairs such that two pairs of the same kind are adjacent is known as a double-slider-crank chain. The following are its inversions:
First inversion (i.e., Elliptical trammel) Link 1 is Fixed
It is an instrument used for drawing ellipses. This inversion is obtained by fixing the slotted plate
Figure 8.7(a) Elliptical trammel
Second inversion (i.e., Scotch yoke): Any of the slider is fixed (i.e. link 2 or link 4) This mechanism is used for converting rotary motion into a reciprocating motion. The inversion is obtained by fixing either the link 2 or link 4. In Fig., link 2 is fixed. In this mechanism, when the link 3 (which corresponds to crank) rotates, the link 1 (which corresponds to a frame) reciprocates. The fixed link 2 guides the frame.
Figure 8.7(b) Scotch yoke
Third inversion (i.e., Actual Oldham’s coupling): Link 3 is fixed
An oldham's coupling is used for connecting two parallel shafts whose axes are at a small distance apart. The shafts are coupled in such a way that if one shaft rotates, the other shaft also rotates at the same speed. This inversion is obtained by fixing the link 3
(b) Worm gears (Non-throated, Single throated, Double throated)
SPUR GEARS: - They have straight teeth parallel to the axes and thus are not subjected to
axial thrust due to teeth load.
Figure 9.1(a) Spur Gear
HELICAL GEARS: - In helical gears, the teeth are curved, each being helical in shape. Two mating gears have the same helix angle, but have teeth of opposite hands. At the beginning of engagement, contact occurs only at the point of leading edge of the curved teeth. As the gears rotate, the contact extends along a diagonal line across the teeth. Thus the load application is gradual which result in now impact stresses and reduction in noise. Therefore, the helical gears can be used at higher velocities then the spur gears and have greater load – carrying capacity.
Figure 9.1(b) Helical Gear
DOUBLE HELICAL AND HERRING BONE GEARS: - A- double- helical gear is equivalent to a pair of helical gears secured together, one having a right – hand helix and the other a left hand helix. The tooth of two raw is separated by a grooved used for too run out. If the left and the right inclinations of a double – helical gear meet at a common apex and there is no groove in between, the gear is known as herring bone gear.
Figure 9.1(c) Double Helical Gear
CROSSED – HELICAL GEARS: - The used of crossed helical gear or spiral gears is limited to light loads. By a suitable choice of helix angle for the mating gears, the two shafts can be set at any angle.
Figure 9.1(d) Crossed Helical Gear
WORM GEARS: - Worm gear is a special case of spiral gear in which the larger wheel, usually, has a hollow or concave shape such that a portion of the pitch diameter is the other gear is enveloped on it. The smaller of two wheels is called the worm which also has larger spiral angle.
Figure 9.1(e) Worm Gear
BEVEL GEARS: - Kinematic ally, the motion between two intersecting shafts is equivalent to the rolling of two cones, assuming no slipping. The gears, in general, are known as bevel gear. When teeth formed on the cones are straight, the gears are known as straight bevel and when inclined, they are known as spiral or helical bevel.
Figure 9.1(f) Bevel Gear
RACK & PINION: - A rack is basically a straight gear used to transmit power and motion in a linear movement.
Figure 9.1(g) Rack & Pinion
Bevel gears are used for the drive to the differential of automobiles.
Spur rack and pinion are used in a lathe.
Helical gears are used for greater load at higher velocities.
Gears are used in different machinery.
AIM: - To study various types of gear trains- simple, compound, reverted and epicyclic
APPARATUS USED: - Gear train system.
A gear train is a combination of gears used to transmit motion from one shaft to another. It becomes necessary when it is required to obtain large speed reduction within a small space.
The following are the main types of gear trains:
Simple gear train.
Compound gear train.
Reverted gear train.
Planetary or Epicyclic gear train.
SIMPLE GEAR TRAIN:
A series of gears, capable of receiving and transmitting motion from one gear to another is called a simple gear train. In it, all the gear axes remain fixed relative to the frame and each gear is on a separate shaft.
When a series of gears are connected in such a way that two or more gears rotate about an axis with the same angular velocity, it is known as compound gear train.
Figure 10.2 Compound Gear Train
If the axes of the first and last wheels of a compound gear coincide; it is called a reverted gear train. Such an arrangement is used in clocks and in simple lathes where ‘back gear’ is used to give a slow speed to the chuck.
Figure 10.3 Reverted Gear Train
PLANETARY OR EPICYCLIC GEAR TRAIN:
When there exists a relative motion of axis in gear train, it is called a planetary or an epicyclic gear train (or simply epicyclic gear or train). Thus in an epicyclic train, the axis of at least one of the gears also moves relative to the frame. Consider two gear wheels A and B, the axis of which are connected by an arm C. if the arm ‘C’ is fixed, the wheels A and B constitute a simple train. However, if the wheel A is fixed so that the arm can rotate about the axis of A, the wheel B would also move around A. therefore, it is an epicyclic train.