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Chapter 4: Problem 18

Determine the time history of the response of a standard mass-spring-damper system when it is subjected to a triangular pulse of magnitude \(F_{0}\) and total duration \(2 \tau\), when the ramp-up and ramp-down times are of the same duration.

Short Answer

Expert verified
Question: Calculate the time-domain response x(t) of a mass-spring-damper system subjected to a triangular force of magnitude F₀ and total duration 2τ, with the ramp-up and ramp-down times being equal. Answer: To find the time-domain response x(t) of the mass-spring-damper system, follow these steps: 1. Define the mass-spring-damper equation of motion: m * (d²x/dt²) + c * (dx/dt) + k * x = F(t) 2. Define the triangular pulse force function F(t). 3. Apply the Laplace transform to the equation of motion. 4. Calculate the Laplace transform of the triangular pulse force F(s). 5. Solve for X(s) in the Laplace domain, assuming the system is initially at rest. 6. Compute the inverse Laplace transform of X(s) to obtain x(t). The final result x(t) represents the displacement of the mass-spring-damper system subjected to the triangular force pulse.

Step by step solution

01

Define the mass-spring-damper equation of motion

The mass-spring-damper system can be modeled by the following second-order linear differential equation of motion: \[m \ddot{x}+c \dot{x}+kx=F(t)\] where - m : mass of the system - c : damping constant - k : spring stiffness - \(x(t)\) : displacement of the system - \(\dot{x}(t)\) : velocity of the system - \(\ddot{x}(t)\) : acceleration of the system - \(F(t)\) : external force acting on the system
02

Define the triangular pulse force

The triangular pulse has a duration of \(2 \tau\) and is divided into two equal parts - the ramp-up and ramp-down. It can be described by the following function: \[F(t)= \begin{cases} F_{0}\frac{t}{\tau} & 0\leq t\leq\tau \\ F_{0}(1-\frac{t-\tau}{\tau}) & \tau<t\leq 2\tau \\ 0 & \text{otherwise} \end{cases} \]
03

Apply Laplace transform to the equation of motion

Using the Laplace transform, we can convert the equation of motion into an algebraic equation. The Laplace transform of the system's equation of motion is: \[m(s^2X(s) - sx(0) - \dot{x}(0)) + c(sX(s) - x(0)) + kX(s) = F(s)\] where \(X(s)\) is the Laplace transform of \(x(t)\), and \(F(s)\) is the Laplace transform of \(F(t)\).
04

Calculate the Laplace transform of the triangular pulse force

To determine the Laplace transform \(F(s)\) of the triangular pulse force, we will calculate the Laplace transforms for both the ramp-up and ramp-down portions of the force, and add them together. The Laplace transforms are: \[F_{1}(s)=\int_{0}^{\tau}F_{0}\frac{t}{\tau}e^{-st} dt = \frac{F_{0}}{\tau}\frac{(1-e^{-\tau s})}{s^2}\] \[F_{2}(s)=\int_{\tau}^{2\tau}F_{0}(1-\frac{t-\tau}{\tau})e^{-st} dt = F_{0}\frac{(e^{-\tau s}-e^{-2\tau s})}{s^2} \] \[F(s)=F_{1}(s)+F_{2}(s)=F_{0}\frac{(1-e^{-2\tau s})}{s^2} \]
05

Solve for X(s) in the Laplace domain

We will now solve for X(s) in the Laplace domain by rearranging the equation of motion: \[X(s)(ms^2 + cs + k) = F(s) + mx(0)s + m\dot{x}(0) + cx(0)\] Assuming that the system is at rest initially (\(x(0)=0\), \(\dot{x}(0)=0\)): \[X(s) = \frac{F(s)}{ms^2 + cs + k} = \frac{F_{0}(1-e^{-2\tau s})}{s^2(ms^2 + cs + k)} \]
06

Perform inverse Laplace transform to obtain x(t)

To obtain the time-domain response x(t), we must compute the inverse Laplace transform of X(s): \[x(t) = \mathcal{L}^{-1}[X(s)]\] Here, the inverse Laplace transform for X(s) may not be easy to obtain directly. One can use a combination of partial fraction decomposition and established Laplace transform tables to determine the inverse Laplace transform for the given system. The final result for x(t) will be a function that represents the displacement of the mass-spring-damper system subjected to the triangular force pulse.

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