Question

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

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

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

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: \[F(t)= \begin{cases} F_{0}\frac{t}{\tau} & 0\leq t\leq\tau \\ F_{0}(1-\frac{t-\tau}{\tau}) & \tau<t\leq 2\tau \\ 0 & \text{otherwise} \end{cases} \]

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 \(F(t)\).

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: \[F_{1}(s)=\int_{0}^{\tau}F_{0}\frac{t}{\tau}e^{-st} dt = \frac{F_{0}}{\tau}\frac{(1-e^{-\tau s})}{s^2}\] \[F_{2}(s)=\int_{\tau}^{2\tau}F_{0}(1-\frac{t-\tau}{\tau})e^{-st} dt = F_{0}\frac{(e^{-\tau s}-e^{-2\tau s})}{s^2} \] \[F(s)=F_{1}(s)+F_{2}(s)=F_{0}\frac{(1-e^{-2\tau s})}{s^2} \]

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 (\(x(0)=0\), \(\dot{x}(0)=0\)): \[X(s) = \frac{F(s)}{ms^2 + cs + k} = \frac{F_{0}(1-e^{-2\tau s})}{s^2(ms^2 + cs + k)} \]

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.

Key concepts

Laplace Transform

The Laplace Transform is a powerful mathematical tool used to convert differential equations, which can be challenging to solve, into algebraic equations, which are typically simpler to handle. It transforms functions from the time domain into the frequency domain. This change simplifies the analysis of systems, especially in engineering. For our typical mass-spring-damper system, the equation of motion given as a second-order differential equation can be transformed into an algebraic equation using the Laplace Transform. This is expressed as:

• The transformed motion equation is: \[m(s^2X(s) - sx(0) - \dot{x}(0)) + c(sX(s) - x(0)) + kX(s) = F(s)\]
• Where \(X(s)\) is the transform of the system displacement \(x(t)\).

This transformation not only helps in simplifying the solution process but also allows one to incorporate initial conditions directly, leading to straightforward solutions in the transformed domain. Once solved for \(X(s)\), you can convert it back to the time domain to find \(x(t)\) using the inverse Laplace Transform.

Differential Equations

Differential equations serve as the mathematical means to represent physical phenomena involving rates of change. In the context of a mass-spring-damper system, the differential equation captures how the system responds over time due to different forces acting upon it. The standard form includes mass (\(m\)), damping constant (\(c\)), spring stiffness (\(k\)), and the external force (\(F(t)\)).

• This formula is: \[m \ddot{x} + c \dot{x} + kx = F(t)\]
• \(\ddot{x}(t)\) is the acceleration, \(\dot{x}(t)\) is the velocity, and \(x(t)\) is the displacement.

To solve this equation, one must often rely on mathematical techniques like the Laplace Transform or numerical methods if an analytical solution proves complex. These techniques allow for deeper insights into how systems behave over time under different force applications.

Triangular Pulse Force

A triangular pulse force is a type of external force characterized by a linear increase to a peak, followed by a linear decrease back to zero. This force pattern can effectively simulate real-world situations where the load ramps up and then decreases symmetrically. For our system:

• The triangular force is defined over a time interval of \(2\tau\), with equal ramp-up and ramp-down durations.
• The force equation is: \[F(t) = \begin{cases} F_{0}\frac{t}{\tau} & 0 \leq t \leq \tau \F_{0}(1 - \frac{t - \tau}{\tau}) & \tau < t \leq 2\tau \0 & \text{otherwise}\end{cases}\]

This setup is particularly useful in mechanical engineering to test how structures respond to variable forces, helping to establish the durability and behavior of the system.

Mechanical Vibrations

Mechanical vibrations involve the oscillation of a mechanical system due to external forces. The mass-spring-damper system is a classic example, often used to study vibration phenomena. Such systems have innate properties:

• The mass (\(m\)) provides inertia.
• The spring stiffness (\(k\)) defines the restoring force.
• The damping constant (\(c\)) dissipates energy, controlling oscillation.

When subjected to forces like the triangular pulse, these systems demonstrate specific responses dependent on parameters like natural frequency and damping ratio. By solving the system's differential equation, typically through methods such as the Laplace Transform, you can observe how the mass's displacement varies over time. This understanding is crucial in predicting how systems behave under different forcing scenarios, enabling engineers to design more stable and reliable products.