Question 5.7: Find the forced response of the undamped single degree of fr...

The forced response of the undamped single degree of freedom system to an arbitrary forcing function is given by
x(t) = \frac{1}{mω} \int_{0}^{1}F(τ) \sin ω(t − τ) dτ
The forcing function F(t) shown in Fig. 5.13 is defined as

F(t) = F_{0}t/t_{1}, 0 ≤ t ≤ t_{1}
= F_{0}, t> t_1
Therefore, the forced response is given by

x(t) = \frac{1}{mω}\left[\int_{0}^{t_1}\frac{F_{0}τ}{t_1} \sin ω(t − τ) dτ + \int_{t_1}^{t}F_{0} \sin ω(t − τ) dτ\right]
Using integration by parts, the response x(t) is given by

=\frac{F_{0}}{mω}\left\{\left.\frac{\cos ω(t − t_{1})}{ω} +\frac{\sin ω(t − τ)}{ω_{2}t_{1}}\right|_0^{t_1} + \frac{1}{ω} − \frac{\cos ω(t − t_1)}{ω}\right\}

= \frac{F_{0}}{mω} \left(\frac{1}{ω} + \frac{\sin ω(t − t_1)}{ω^{2}t_{1}} − \frac{\sin ωt}{ω^{2}t_{1}}\right)