A 128-lb weight is attached to a spring having a spring constant of 64 lb/ft. The weight is started in motion with no initial velocity by displacing it 6 in above the equilibrium position and by simultaneously applying to the weight an external force F(t) = 8 sin 4t. Assuming no air resistance,

Question

A 128-lb weight is attached to a spring having a spring constant of 64 lb/ft. The weight is started in motion with no initial velocity by displacing it 6 in above the equilibrium position and by simultaneously applying to the weight an external force F(t) = 8 sin 4t. Assuming no air resistance, find the subsequent motion of the weight.

Accepted Answer

Here [latex]m=4, k=64, a=0, ext { and } F(t)=8 \sin 4 t[/latex]; hence, Eq. (14.1) becomes

[latex]\ddot{x}+16 x=2 \sin 4 t[/latex] (1)

This problem is, therefore, an example of forced undamped motion. The solution to the associated homogeneous equation is

[latex]x_{h}=c_{1} \cos 4 t+c_{2} \sin 4 t[/latex]

A particular solution is found by the method of undetermined coefficients (the modification described in Chapter 11 is necessary here): [latex]x_{p}=-\frac{1}{4} \cos 4 t[/latex]. The solution to (1) is then

[latex]x=c_{1} \cos 4 t+c_{2} \sin 4 t-\frac{1}{4} t \cos 4 t[/latex]

Applying the initial conditions, [latex]x(0)=-\frac{1}{2} ext { and } \dot{x}(0)=0[/latex] , we obtain

[latex]x=-\frac{1}{2} \cos 4 t+\frac{1}{16} \sin 4 t-\frac{1}{4} t \cos 4 t[/latex]

Note that [latex]|x| ightarrow \infty ext { as } t ightarrow \infty[/latex]. This phenomenon is called pure resonance. It is due to the forcing function F(t) having the same circular frequency as that of the associated free undamped system.

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Eq. (14.1) : [latex]\ddot{x}+\frac{a}{m} \dot{x}+\frac{k}{m} x=\frac{F(t)}{m}[/latex]

Question 14.10: A 128-lb weight is attached to a spring having a spring cons...

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