How do engineers determine the weight of an airplane which they are designing? How do engineers determine the location of the center of gravity for an aircraft which they are designing? How do engineers determine the weight of an airplane which they are designing? Weight is the force generated by the gravitational attraction of the earth on the airplane. Each part of the aircraft has a unique weight and mass, and for some problems it is important to know the distribution. But for total aircraft maneuvering, we only need to be concerned with the total weight and the location of the center of gravity. The center of gravity is the average location of the mass of any object. How do engineers determine the weight of an airplane which they are designing? An airplane is a combination of many parts; the wings, engines, fuselage, and tail, plus the payload and the fuel. Each part has a weight associated with it which the engineer can estimate, or calculate, using Newton's weight equation: w = m * g where w is the weight, m is the mass, and g is the gravitational constant which is 32.2 ft/square sec in English units and 9.8 meters/square sec in metric units. The mass of an individual component can be calculated if we know the size of the component and its chemical composition. Every material (iron, plastic, aluminum, gasoline, etc.) has a unique density. Density r is defined to be the mass divided by the volume v: r = m / v If we can calculate the volume m = r * v The total weight W of the aircraft is simply the sum of the weight of all of the individual components. W = w(fuselage) + w(wing) + w(engines) + w(payload) + w(fuel) + ... To generalize, if we have a total of "n" discrete components, the weight of the aircraft is the sum of the individual i component weights with the index i going from 1 to n. The greek mathematical symbol sigma is used by mathematicians to denote this addition. (Sigma is a zig-zag symbol with the index designation being placed below the bottom bar, the total number of additions placed over the top bar, and the variable to be summed placed to the right of the sigma with each component designated by the index.) W = SUM(i=1 to i=n) [wi] This equation says that the weight of the airplane is equal to the sum of the weight of "n" discrete parts. What if the parts are not discrete? What if we had a continuous change of mass from front to back? The continuous change can be computed using integral calculus. The sigma designation is changed to the integral "S" shaped symbol to denote a continuous variation. W = S w(x) dx The discrete weight is replaced with w(x) which indicates that the weight is some function of distance x. If we are given the form of the function, there are methods to solve the integration. If we don't know the actual functional form, we can still numerically integrate the equation using a spread sheet by dividing the distance up into a number of small distance segments and determining the average value of the weight over that small segment, then summing up the value. How do engineers determine the location of the center of gravity for an aircraft which they are designing? In general, determining the center of gravity (cg) is a complicated procedure because the mass (and weight) may not be uniformly distributed throughout the object. The general case requires the use of calculus which we will discuss at the bottom of this page. If the mass is uniformly distributed, the problem is greatly simplified. If the object has a line (or plane) of symmetry, the cg lies on the line of symmetry. For a solid block of uniform material, the center of gravity is simply at the average location of the physical dimensions. (For a rectangular block, 50 X 20 X 10, the center of gravity is at the point (25,10, 5) ). For a triangle of height h, the cg is at h/3, and for a semi-circle of radius r, the cg is at (4*r/(3*pi)) where pi is ratio of the circumference of the circle to the diameter. There are tables of the location of the center of gravity for many simple shapes in math and science books. The tables were generated by using the equation from calculus shown on the slide. For a general shaped object, there is a simple mechanical way to determine the center of gravity: 1. If we just balance the object using a string or an edge, the point at which the object is balanced is the center of gravity. (Just like balancing a pencil on your finger!) 2. Another, more complicated way, is a two step method shown on the slide. In Step 1, you hang the object from any point and you drop a weighted string from the same point. Draw a line on the object along the string. For Step 2, repeat the procedure from another point on the object You now have two lines drawn on the object which intersect. The center of gravity is the point where the lines intersect. This procedure works well for irregularly shaped objects that are hard to balance. If the mass of the object is not uniformly distributed, we must use calculus to determine center of gravity. We will use the symbol S dw to denote the integration of a continuous function with respect to weight. Then the center of gravity can be determined from: cg * W = S x dw where x is the distance from a reference line, dw is an increment of weight, and W is the total weight of the object. To evaluate the right side, we have to determine how the weight varies geometrically. From the weight equation, we know that: w = m * g where m is the mass of the object, and g is the gravitational constant. In turn, the mass m of any object is equal to the density, rho, of the object times the volume, V: m = rho * V We can combine the last two equations: w = g * rho * V then dw = g * rho * dV dw = g * rho(x,y,z) * dx dy dz If we have a functional form for the mass distribution, we can solve the equation for the center of gravity: cg * W = g * SSS x * rho(x,y,z) dx dy dz where SSS indicates a triple integral over dx. dy. and dz. If we don't know the functional form of the mass distribution, we can numerically integrate the equation using a spreadsheet. Divide the distance into a number of small volume segments and determining the average value of the weight/volume (density times gravity) over that small segment. Taking the sum of the average value of the weight/volume times the distance times the volume segment divided by the weight will produce the center of gravity.