Figure 3.11(a) shows a pendulum which is supported by two springs which have stiffness coefficients k1 and k2. The two springs are connected to the pendulum rod at points which are at distances a and b from the pin joint, as shown in the figure. If the two springs shown in Fig. 3.11(a) are to be

Question

Figure 3.11(a) shows a pendulum which is supported by two springs which have stiffness coefficients [latex]k_{1}[/latex] and [latex]k_{2}[/latex]. The two springs are connected to the pendulum rod at points which are at distances a and b from the pin joint, as shown in the figure. If the two springs shown in Fig. 3.11(a) are to be equivalent to a single spring which is connected to the rod at a distance d from the pin joint as shown in the figure, determine the stiffness coefficient of this single spring in terms of the constants [latex]k_{1}[/latex] and [latex]k_{2}[/latex].

Accepted Answer

As shown in Fig. 3.11(c), the force produced by the springs [latex]k_{1}[/latex] and [latex]k_{2}[/latex], as the results of an angular displacement θ, are given by

[latex]F_{1} = −k_{1}[/latex]a sin θ, [latex]F_{2} = −k_{2}[/latex]b sin θ

The resultant moment due to the forces [latex]F_{1}[/latex] and [latex]F_{2}[/latex] about point O is given by
M = [latex]F_{1}[/latex]a cos θ + [latex]F_{2}[/latex]b cos θ = −[latex](k_{1}a² + k_{2}[/latex]b²) sin θ cos θ

The force [latex]F_{e}[/latex] produced by the equivalent spring is given by

[latex]F_{e} = −k_{e}[/latex]d sin θ

and the moment about O produced by this force is
[latex]M_{e} = F_{e}[/latex]d cos θ = −[latex]k_{e}[/latex]d² sin θ cos θ

If the two systems in Fig. 3.11(a) and (b) are to be equivalent, the moment M produced by the original system must be equal to the moment Me produced by the equivalent system, that is, M = [latex]M_{e}[/latex], or
[latex]−(k_{1}a² + k_{2}b²)[/latex] sin θ cos θ = −[latex]k_{e}d²[/latex] sin θ cos θ

from which the equivalent stiffness coefficient [latex]k_{e}[/latex] is defined as
[latex]k_{e} = \frac{k_{1}a² + k_{2}b²}{d²}[/latex]
A moment equation is used in this example instead of a force equation to determine [latex]k_{e}[/latex], because this is the case of angular oscillations in which the system degree of freedom is the angular orientation θ.

Question 3.4: Figure 3.11(a) shows a pendulum which is supported by two sp...

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