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Chapter 4: Problem 19

Determine the shock spectrum for a linear system subjected to the transition loading of Section 4.6.2.

Short Answer

Expert verified
In summary, to determine the shock spectrum for a linear system subjected to the transition loading of Section 4.6.2, the following steps have to be undertaken: 1. Derive the equation of motion for the system; 2. Express the given transition loading; 3. Calculate the response of the system using the equation of motion and the given force; 4. Calculate the shock response spectrum by evaluating the maximum response of the system for various natural frequencies; 5. Interpret the resulting shock response spectrum to understand the system's behavior during the transient event and design the system accordingly. To show the most essential parts of the solution, focus on the response calculation (Step 3) and the process of determining the shock response spectrum (Step 4).

Step by step solution

01

Equation of motion

For a linear system such as this, the equation of motion can be expressed as: \[ m\ddot{u} + c\dot{u} + ku = F(t) \] where: \(m\) = mass of the system, \(c\) = damping coefficient, \(k\) = stiffness of the system, \(u\) = displacement, \(F(t)\) = external force acting on the system, which is given as the transition loading.
02

Transition loading

From Section 4.6.2, the transition loading can be expressed as: \[ F(t) = F_0 \left(1 - e^{-\frac{t}{T}} \right) \] where: \(F_0\) = maximum magnitude of the force, \(T\) = time constant of the transition loading.
03

Response of the system

To determine the response of the system (\(u(t)\)), we need to solve the equation of motion with the given force, \(F(t)\). This can be done using the Laplace transform method or other numerical methods. The general solution for this equation can be given as: \[ u(t) = A_0 e^{-\omega_d t}cos(\omega_n t - \varphi) \] where: \(A_0\) = amplitude of the response, \(\omega_d\) = damped natural frequency, \(\omega_n\) = undamped natural frequency, \(\varphi\) = phase angle.
04

Calculate the shock response spectrum

The shock response spectrum (SRS) can be calculated by evaluating the maximum response of the system for various natural frequencies. Using the equation from Step 3, we can calculate the maximum response, \(u_{max}\), as a function of the natural frequency (\(\omega_n\)): \[u_{max} (\omega_n) = \frac{F_0}{k} \sqrt{1 + \left(2 \zeta \frac{\omega_n}{\omega_0}\right)^2 } \] where: \(\omega_0\) = reference natural frequency, \(\zeta\) = damping ratio. By varying the natural frequency, we can plot the shock response spectrum for the given transition loading.
05

Interpret the results

The shock spectrum obtained represents the maximum dynamic displacement of the system subjected to the transition loading. It helps in understanding how the system responds to a transient event and assists in designing the system to withstand such forces. Analyzing the spectrum can also provide insight into frequency regions when the system exhibits excessive response, which can guide further optimization to control the response during dynamic events.

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Most popular questions from this chapter

The timing device of Problem \(2.19\) is tapped to initiate motion. Determine the magnitude of the impulse required so that the motion of the device has an amplitude \(\Theta_{0}\).

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