An electric overhead traveling crane, consisting of a girder, trolley, and wire rope, is shown in figure. The girder has a flexural rigidity (EI) of 6x10^{12} lb.in^{2} and a span (L) of 30 ft. The rope is made of steel and has a length of 20 ft. The weights of the trolley and the load lifted are 8000lb and 2000lb, respectively.
1. Find and compute the natural frequencies, mode ratios, normal modes and sketch mode shape diagrams for the normal modes.
2. Use Matlab to simulate the response of the system to initial excitations that will excite the normal modes:
Case 1: mode 1 use x_{2}(0) = 0.0003m,?_{2}(0)=0, x_{1}(0)=x_{2}(0)/r_{1},?_{1}(0)=0
Case 2: mode 2 use x_{2}(0) = 0.03m,?_{2}(0)=0,x1(0) =x_{2}(0)/r2,?_{1}(0)=0
Note: these I.C. conform to the respective normal modes.
Compare the nature of computational responses from Matlab to the theoretical predicts.
3. Use Matlab to simulate the response of the system to a set of general initial conditions that do not conform to the mode ratios:
Case 3: general I.C. x_{2}(0)= 0.0003m,?_{2}(0)=0,x_{1}(0) =0,?_{1}(0)=0
Hint: For general initial conditions, the response conforms to the principle of superposition of the natural modes.