Question 3.6.1: Calculate the response spectrum for the forcing function giv......

As in Example 3.2.2, the forcing function F(t) sketched in Figure 3.16 can be written as the sum of two other simple functions. In this case, the input is the sum of

F_1(t)=\frac{t}{t_1} F_0

and

F_2(t)= \begin{cases}0 & 0<t<t_1 \\ -\frac{t-t_1}{t_1} F_0 & t \geq t_1\end{cases}

Following the steps taken in Example 3.2.2, the response is calculated by evaluating the response to F_1(t) and separately to F_2(t). Linearity is then used to obtain the total response to F(t)=F_1(t)+F_2(t). The response to F_1(t), denoted by x_1(t), calculated using equations (3.71) and (3.73), becomes

x(t)=\int_0^t F(\tau) h(t-\tau) d \tau           (3.71)

h(t-\tau)=\frac{1}{m \omega_n} \sin \omega_n(t-\tau)          (3.73)

x_1(t)=\frac{\omega_n}{k} \int_0^t \frac{F_0 \tau}{t_1} \sin \omega_n(t-\tau) d \tau=\frac{F_0}{k}\left(\frac{t}{t_1}-\frac{\sin \omega_n t}{\omega_n t_1}\right)       (3.75)

Similarly, the response to F_2(t), denoted by x_2(t), becomes

x_2(t)=\int_0^t F_2(\tau) \frac{1}{m \omega_n} \sin \omega_n(t-\tau) d \tau=\frac{-F_0}{m \omega_n} \int_{t_1}^t \frac{\tau-t_1}{t_1} \sin \omega_n(t-\tau) d \tau

which becomes

x_2(t)=-\frac{F_0}{k}\left[\frac{t-t_1}{t_1}-\frac{\sin \omega_n\left(t-t_1\right)}{\omega_n t_1}\right]         (3.76)

so that the total response becomes the sum x(t)=x_1(t)+x_2(t) :

x(t)= \begin{cases}\frac{F_0}{k}\left(\frac{t}{t_1}-\frac{\sin \omega_n t}{\omega_n t_1}\right) & t<t_1 \\ \frac{F_0}{k \omega_n t_1}\left[\omega_n t_1-\sin \omega_n t+\sin \omega_n\left(t-t_1\right)\right] & t \geq t_1\end{cases}       (3.77)

Alternately, the Heaviside step function may be used to write this solution as

x(t)=\frac{F_0}{k}\left(\frac{t}{t_1}-\frac{\sin \omega_n t}{\omega_n t_1}\right)-\frac{F_0}{k}\left(\frac{t-t_1}{t_1}-\frac{\sin \omega_n\left(t-t_1\right)}{\omega_n t_1}\right) \Phi\left(t-t_1\right)         (3.78)

Equation (3.77) is the response of an undamped system to the excitation of Figure 3.16. To find the maximum response, the derivative of equation (3.77) is set equal to zero and solved for the time t_p at which the maximum occurs. This time t_p is then substituted into the response x\left(t_p\right) given by equation (3.77) to yield the maximum response x\left(t_p\right). Differentiating equation (3.77) for t>t_1 yields \dot{x}\left(t_p\right)=0 or

-\cos \omega_n t_p+\cos \omega_n\left(t_p-t_1\right)=0        (3.79)

Using simple trigonometry formulas and solving for \omega_n t_p yields

\tan \omega_n t_p=\frac{1-\cos \omega_n t_1}{\sin \omega_n t_1} \quad \text { or } \quad \omega_n t_p=\tan ^{-1}\left(\frac{1-\cos \omega_n t_1}{\sin \omega_n t_1}\right)        (3.80)

where t_p denotes the time to the first peak [i.e., the time for which the maximum value of equation (3.77) occurs]. Expression (3.80) corresponds to a right triangle of sides \left(1-\cos \omega_n t_1\right), and \sin \omega_n t_1, and hypotenuse

\sqrt{\sin ^2 \omega_n t_1+\left(1-\cos \omega_n t_1\right)^2}=\sqrt{2\left(1-\cos \omega_n t_1\right)}       (3.81)

This relationship is illustrated in Figure 3.17. Hence, \sin \omega_n t_p can be calculated from

\sin \omega_n t_p=-\sqrt{\frac{1}{2}\left(1-\cos \omega_n t_1\right)}          (3.82)

and

\cos \omega_n t_p=\frac{-\sin \omega_n t_1}{\sqrt{2\left(1-\cos \omega_n t_1\right)}}            (3.83)

Substitution of this expression into solution (3.77) evaluated at t_p yields, after some manipulation [here \left.x\left(t_p\right)=x_{\max }\right],

\frac{x_{\max } k}{F_0}=1+\frac{1}{\omega_n t_1} \sqrt{2\left(1-\cos \omega_n t_1\right)}         (3.84)

where the left side represents the dimensionless maximum displacement. It is customary to plot the response spectrum (dimensionless) versus the dimensionless frequency

\frac{t_1}{T}=\frac{\omega_n t_1}{2 \pi}          (3.85)

where T is the structure’s natural period. This provides a scale related to the characteristic time, t_1, of the input. Figure 3.18 is a plot of the response spectrum for the ramp input force of Figure 3.16. Note that each point on the plot corresponds to a different rise time, t_1, of the excitation. The vertical scale is an indication of the relationship between the structure and the rise time of the excitation.

The response is plotted using equation (3.77) along with the maximum magnitude as given by equation (3.84) and the ramp input function in Figure 3.19. Note from these plots that the amplitude of the response is magnified, or larger than the level of the input force. If t_1 is chosen to be near a period (the minimum in Figure 3.18), then the response is lower than this value and the maximum response will be equal to the input level. The effects of the various parameters form the topic of shock isolation (Section 5.2, which can be read now) and are examined numerically in Section 3.9.