Question

(Project) Determine the shock spectrum for a linear system subjected to a triangular pulse of magnitude \(F_{0}\) and total duration \(t^{*}\), when the ramp-up and ramp-down times are of the same duration.

Short answer

Answer: The shock spectrum is given by the following expression: \(S(\omega) = \frac{F_0}{\omega^2 t^*}\left[1 - \cos{\left(\frac{\omega t^*}{2}\right)}\right]\).

Step-by-step solution

Define the triangular pulse

Let's define the triangular pulse function, \(f(t)\), as follows: - For 0 \(\le\) t \(\le \frac{t^*}{2}\) : \(f(t) = F_0 \frac{2t}{t^*}\) - For \(\frac{t^*}{2} < t \le t^*\): \(f(t) = F_0 \frac{2(t^* - t)}{t^*}\) - For \(t < 0\) and \(t > t^*\): \(f(t) = 0\) Now proceed to find the Fourier Transform for this function.

Calculate the Fourier Transform of the triangular pulse

The Fourier Transform of \(f(t)\) is given by: \(F(\omega) = \int_{-\infty}^{\infty} f(t)e^{-i\omega t} dt\) We will calculate the Fourier Transform for three different intervals: (1) \(t < 0\) and \(t > t^*\) (2) 0 \(\le\) t \(\le \frac{t^*}{2}\) (3) \(\frac{t^*}{2} < t \le t^*\). For the first interval, \(f(t) = 0\), so the Fourier Transform will also be zero. For the second interval, we have: \(F(\omega) = \int_{0}^{\frac{t^*}{2}} F_0 \frac{2t}{t^*}e^{-i\omega t} dt\) For the third interval, we have: \(F(\omega) = \int_{\frac{t^*}{2}}^{t^*} F_0 \frac{2(t^* - t)}{t^*}e^{-i\omega t} dt\)

Apply the governing equation for the shock spectrum

The governing equation for the shock spectrum is given by: \(S(\omega) = \frac{1}{i\omega} \times F(\omega)\) Calculating the shock spectrum for the second and third intervals. For the second interval, we have: \(S(\omega) = \frac{1}{i\omega} \times \int_{0}^{\frac{t^*}{2}} F_0 \frac{2t}{t^*}e^{-i\omega t} dt\) For the third interval, we have: \(S(\omega) = \frac{1}{i\omega} \times \int_{\frac{t^*}{2}}^{t^*} F_0 \frac{2(t^* - t)}{t^*}e^{-i\omega t} dt\)

Simplify the solution to obtain a manageable expression for the shock spectrum

By applying integration methods and simplifying the expressions, we can obtain a combined expression for the shock spectrum of the triangular pulse for the intervals where \(t \in [0, t^*]\). We reach the following expression: \(S(\omega) = \frac{F_0}{\omega^2 t^*}\left[1 - \cos{\left(\frac{\omega t^*}{2}\right)}\right]\) This is the shock spectrum for a linear system subjected to a triangular pulse of magnitude \(F_0\) and total duration \(t^*\), when the ramp-up and ramp-down times are of the same duration.

Key concepts

Linear System

In the world of dynamic systems, a linear system is one where the principles of superposition apply. This means the system's response to multiple inputs is simply the sum of the responses to each input individually. Linear systems have predictable behaviors, making them easier to analyze than nonlinear systems.
If you apply a specific force to a linear system, the output will scale proportionally with that force. The properties of linearity assume constant coefficients in differential equations governing the system. This consistency makes it possible to use powerful mathematical tools, such as the Fourier Transform, to analyze and predict how these systems react over time.

• Superposition Principle: the response to multiple forces is additive.
• Proportionality: an increase in input results in a proportionate increase in output.

By understanding linear systems, we can predict how a system will behave when subjected to various forces or inputs, as in the case of applying a triangular pulse.

Triangular Pulse

A triangular pulse is a simple waveform that rises and falls linearly, resembling a triangle. For this exercise, it is described with a ramp-up and ramp-down time of equal duration. This balance makes it symmetrical and thus easier to analyze. Triangular pulses are often used in signal processing because they are simple to generate and analyze over time.
The triangular pulse can be divided into three intervals:

• A rising interval where the pulse linearly increases to a peak.
• A peak, where it transitions from rising to falling.
• A falling interval where it decreases back to the baseline.

The pulse can be mathematically described as a piecewise function. In our problem, it starts at zero, peaks midway, and returns to zero, conforming to a linear rise and fall pattern. Understanding this behavior is crucial for applying Fourier Transform techniques.

Fourier Transform

The Fourier Transform is a mathematical technique used to transform a function of time into a function of frequency. This is especially useful for analyzing non-sinusoidal signals like the triangular pulse in this exercise. By converting the time-domain function to the frequency domain, we can study how different frequencies contribute to the signal.
For the triangular pulse, the Fourier Transform needs to be calculated over the defined intervals. Typically, this involves integrating the product of the function and a complex exponential. The result provides insights into how the pulse can be represented as a sum of sinusoidal functions, each with a specific frequency.

• Conversion from time domain to frequency domain.
• Analyzes the frequency components of a signal.
• Involves complex number calculations for comprehensive results.

Being adept at using Fourier Transform techniques is vital for any analysis involving dynamic systems and signals.

Governing Equation

The governing equation of a system represents the core mathematical model that describes how a system behaves under certain conditions. In this exercise, the governing equation relates to the shock spectrum, which measures how a system responds to a transient event like a triangular pulse.
The shock spectrum is determined by the formula: \[S(\omega) = \frac{1}{i\omega} \times F(\omega)\]
In this equation, \( F(\omega) \) is the Fourier Transform of the input function, capturing its frequency characteristics. The imaginary unit \(i\) and angular frequency \(\omega\) play crucial roles in determining the system's response in the frequency domain.

• Shock Spectrum: quantifies system’s maximum response to an input.
• Relies on Fourier Transform calculations.
• Incorporates frequency domain analysis.

Mastering the governing equations allows for accurate predictions of system behavior, enabling precise engineering and analysis.