 Hello students, I am Dr. Bhargesh Deshmukh, Professor in Mechanical Engineering Department, Vahalchand Institute of Technology, Solapur. This session is on design of lever, this is the topic from the course Machine Design 1 and the end of this session we will be able to establish the design equations for a lever. Let us see how a lever is to be designed. The lever length is decided as per the leverage required. If you have a long lever we can lift heavy load with the help of a small effort. We can see it in the tommy bar of a wheel, when we want to remove the wheel of a truck usually the cleaner or the driver uses a long lever and usually they stand on it to remove. This is what is the leverage, with the help of a small force, long lever length we can apply a great torque, great force and that is what is the beauty of a lever. Lever length is very important in the design of a lever. Then we need to lift a load F by means of an effort P. We need to design the lever accordingly and select the leverage. The cross section of the lever is designed on the basis of bending stresses. We need to consider the bending failure which is a dominant failure for a lever cross section. In this design of lever the first step is to calculate the forces. Force analysis becomes very important. On this lever these two are the forces, the load F and the effort P, there is the fulcrum distance of the load is L2 from the fulcrum and distance of effort is L1 from the fulcrum. The load is taken as F and the effort P is required to produce the force is calculated by taking the moments at about the fulcrum. This is the fulcrum point where we are supposed to take the moments after. We can use the equation F equals L2 equals P into L1. We can use this and calculate the required force. The required force is P which is equal to F multiplied by L2 by L1. L2 by L1 are the dimensions of the lever. This is the first type of the lever. Then the vertical force denoted is the reaction R. We are supposed to calculate this reaction R also. The vertical forces acting on the lever must be in equilibrium. In order to keep this system in equilibrium, F, P, the summation must be equal to this vertical force R, vertical upward force R as F and P are vertically downwards. Therefore, R equals F plus P. This is the equation that we need to use. This is the case where in forces are parallel on the lever. There can be a case where we can find inclined forces. Let us see the second type of the lever, F is in between R and P. The load and the effort act in opposite direction. This is the load upward vertically and P effort vertically downward. These are opposite in nature. Vertical forces acting on the lever must be in equilibrium. Same equation we need to use. F equals R plus P that is the change equation. We need to keep this system in equilibrium. This is again the case where forces are parallel. Then next, if the forces acting on the lever are inclined, they are not straight vertically upward or downward. But those are inclined to the axis of the lever. We need to analyze with a somewhat different methodology. L1 is the perpendicular distance for line of action of force P. And L2 is the perpendicular distance for line of action of force F. These forces are inclined, not parallel. Analysis become slightly different. For this bell crank lever, the arms are inclined at an angle theta. The force F and P act right angle to the arm line. This is the arm line of the arm. Force F is perpendicular to it. This is the other arm and effort P is perpendicular to it. Then L1 and L2 are the corresponding arm lengths. Reaction at the fulcrum is equal to the resultant of F and P. F and P, these two, the reaction at fulcrum is the resultant of F and P. In which equation is to be used? You can think upon it. The line of action R passes through the intersection of P and F. Line of action of R is this. It should pass through point O, which is the intersection of extension of line F and P. You can recall from where this formula has been used. R equals F square plus P square minus 2 Fp cos theta square root of it. For a right angle bell crank lever, where this theta becomes 90, cos theta equals 0. The formula can be simplified. R equals square root of F square plus P square. Recall the concept of parallelogram of forces and law of parallelogram of forces. Then, next step is finding out the lever cross section. Lever cross section is subjected to bending moment for a two-arm lever. This F and P are acting over here. This is the boss section. L1 is the length, effort of L1 minus D1. It is the weak section where the lever may fail or it is prone for failure. Bending moment is 0 at the point of application of P and RF and maximum at the boss section. This is the boss zone. Here, the bending moment will be maximum. The lever cross section can be either rectangular or analytic. Accordingly, we need to select the equations for section models. Now, in this lever, bending moment is maximum at the section axis, which is given by mB equals P. P is the effort P. L1 minus this D because total diameter, outside diameter of the boss is 2D. This is D. L1 minus D is the momentum. Cross section of the lever can be rectangular, elliptical or I. Let us take in case of rectangular cross section. For a rectangular cross section, I equals BD cube by 12 and Y becomes D by 2. Where B is parallel to the neutral axis, B and D is perpendicular to the neutral axis. And the condition is B equals 2 times B. That is what is the assumption we need to do. If the cross section of the lever is an elliptical, then I changes to pi BA cube by 64 and Y becomes A by 2. What is A and B? Where A is the major axis and B is the minor axis and the condition is A equals twice of B. The dimensions of the lever cross section can be obtained by the equation sigma B equals mB Y by I. This is nothing but bending stress equals m by I by Y, where I by Y is Z of the section models. We need to take a great care while selecting the section, cross section of a lever. In the previous slide, we have seen that the bending moment is maximum at the fulcrum point. And it is minimum at or it is 0 at the point of application of effort. It is very clear that bending moment is minimum and maximum. Therefore, it is not advisable to use uniform cross section of a lever. Therefore, what we can do is we can use a tapered lever in order to save the material. It is called as lever of uniform strength. We have seen this part. The final equation we have seen for rectangular cross section. And then for an elliptical cross section bending moment, the equation. This is what is the design of a lever. For a fulcrum pin, we need to use the bearing consideration. D1 and L1 are the pin dimensions. We need to use bearing pressure for the pin design. The projected area is D1 L1. R is equal to bearing pressure multiplied by the projected area. P is the permissible bearing pressure. The length and diameter ratio is taken as 1 is to 2. Ode of the boss is taken as 2 times the pin diameter. Phosphor-bronze-boshock 3 mm thick is used to reduce the friction. Permissible bearing pressure is around 5 to 10 Newton per Newton square.