Three identical springs, each of stiffness k, support a block of mass m, as shown in Fig. (a). Deformation of bar AB may be neglected. (1) Find the equivalent spring stiffness k0—that is, the stiffness of a single spring as shown in Fig. (c)—that can replace the original springs without changing

Question

Three identical springs, each of stiffness k, support a block of mass m, as shown in Fig. (a). Deformation of bar AB may be neglected. (1) Find the equivalent spring stiffness [latex]k_0[/latex]—that is, the stiffness of a single spring as shown in Fig. (c)—that can replace the original springs without changing the displacement characteristics of the block. (2) If the block mass 0.2 kg and k = 875 N/m, find the circular frequency, the frequency, and the period of free vibration.

Accepted Answer

Part 1

Figure (b) shows a method for calculating the equivalent spring stiffness. First, a static vertical force F is applied at point D, and the vertical movement [latex]δ_D[/latex] is computed. The equivalent spring stiffness [latex]k_0[/latex] is then calculated using [latex]F = k_0δ_D[/latex]. Following this procedure, the mass in Fig. (c) will have the same displacement characteristics as the mass in Fig. (a).
From equilibrium analysis of Fig. (b), we see that the spring forces are equal to F in the lower spring and F/2 in each of the upper springs (C is the midpoint of bar AB). The elongations of the upper springs are, therefore, identical, and the vertical displacements [latex]δ_A, δ_B[/latex], and [latex]δ_C[/latex] are each equal to (F/2)/k (note that bar AB remains horizontal). Next we observe that the vertical movement of D equals the vertical movement of C plus the elongation of the lower spring; that is, [latex]δ_D = δ_C + (F/k)[/latex]. The equivalent spring stiffness thus becomes

[latex]k_0 = \frac{F}{δ_D}= \frac{F}{δ_C + (F/k)} = \frac{F}{(F/2k) + (F/k)}[/latex]
[latex]= \frac{1}{(1/2k) + (1/k)} = \frac{2k}{3}[/latex]

Part 2

Using the given data and the equivalent system shown in Fig. (c), the circular frequency, frequency, and period of free vibration are

[latex]p =\sqrt{\frac{k_0}{m}} =\sqrt{\frac{2k/3}{m}} =\sqrt{\frac{2(875)/3}{0.2}} =[/latex] 54 rad/s

[latex]f = \frac{p}{2π} =\frac{54}{2π} =[/latex] 8.6 Hz

[latex]τ = \frac{1}{f} = \frac{1}{8.6} =[/latex] 0.116 s

Question 20.1: Three identical springs, each of stiffness k, support a bloc...

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