| Title |
Application of Initial Conditions in Beam-realated Differential Equations using Generalized Functions |
| Authors |
Kwak Soon-Seop ; Song Kil-Ho ; Choi Seok-Woo |
| Keywords |
Initial Condition ; Boundary Condition ; Generalized Function ; Delta Function and its Derivatives ; Governing Differential Equations ; Particular Solution ; Double Moment ; Double Shear ; Unit Force ; Unit Moment |
| Abstract |
In Ly(x)=f(x), where L is a liner differential operator, the general solution is y=yh+yp in which yh is a homogeneous solution and yp is a particular solution. The homogeneous solution yh is usually expressed in the form of y=4∑n=1 AnPn(x), where An is constant. Specially when the differential equation is related to the problems of beams, most cases of the problem are when is 4. Then yh is in the form of yh=A1P1(x)+A2P2(x)+A3P3(x)+A4Pa(x), where Pn(x) is to be found. Instead of that forms, yh can be expressed in different form using initial conditions ; y(0),y'(0), y"(0), y"'(0) like as yh=y(0)Ф1(x)+y'(0)Ф2(x)+y"(0)Ф3(x)+y"'(0)Ф4(x). Those initial conditions can also be interpreted as deflection, angle, moment, and shear force at the initial point. The advantages of using initial conditions in the homogeneous solution; ① In most of cases, two of the y(0),y'(0), y"(0) and y"'(0) are zero, so yh has only two terms. Therefore it becomes easier to handle the problem. ② It is usually very tedious and difficult to find the particular solution yp. But when the homogeneous solution is in the form of yh=y(0)Ф1(x)+y'(0)Ф2(x)+y"(0)Ф3(x)+y"'(0)Ф4(x), the particular solution can be easily obtained like as yp=∫0~x f(ξ)Ф4(x-ξ)dξ, where f(ξ) is related to the loads. In other words, the last term Ф4(x) is related to the particular solution. ③ Even though, Ф4(x) is known, the integration of yp is also difficult to manage. But using the characteristics of generalized functions, the load related term f(x) in differential equations can be simply described in the form of continuous function. And yp can be found in easy way. |