1 answer

Problem 3: Find the natural frequency of the system shown in Figure 3. Problem 4: In...

Question:

Problem 3: Find the natural frequency of the system shown in Figure 3. Problem 4: In the mechanical system shown in Figure 4,
Problem 3: Find the natural frequency of the system shown in Figure 3. Problem 4: In the mechanical system shown in Figure 4, assume that the rod is massless, perfectly rigid, and pivoted at point P. The displacement x is measured from the equilibrium position. Assuming that x is small, that the weight mg at the end of the rod is 5 N, and that the spring constant k is 400 N/m, find the natural frequency of the system. 2a 2 ng ky Figure 3: Mechanical System Figure 4: Mechanical System

Answers

Problem 3.

From the figure, it is evident that all the springs K1 , K2 and K3 are connected in parallel, hence equivalent spring constant :

Keq = K1 +K2 + K3

Now,

mddot{x} = -K_{eq}x

ddot{x} = -rac{K_{eq}}{m}x

ddot{x} = -omega ^2x

Kc A,

Problem 4.

By applying energy method,

KE + PE = Constant

dt

KE = rac{1}{2}mdot{x}^2

PE = rac{1}{2}k(x/2)^2

dt、2

mdot{x}ddot{x} + rac{1}{4}kxdot{x} = 0

dot{x}(mddot{x} + rac{1}{4}kx) = 0

rメ0

4m

4m

mg = 5 N

Taking g = 10 m/s2.

m = 5/10 = 0.5 kg

k = 400 N/m

400 = V200 = 14.142Hz 4 0.5

.

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