Question

Determine the response of a standard mass-spring-damper system when it is subjected to a sinusoidal transition of duration \(t^{*}\) to a constant load of magnitude \(F_{0}\).

Short answer

Question: Determine the response of a mass-spring-damper system to a sinusoidal transition of duration \(t^{*}\) to a constant load of magnitude \(F_{0}\). Assume the system is initially at rest. Answer: To determine the response of the mass-spring-damper system, follow these steps: 1. Write down the equation of motion for the system. 2. Define the force function for the sinusoidal transition and constant load. 3. Find the homogeneous and particular solutions of the equation of motion. 4. Determine the nature of the homogeneous solution using the characteristic equation and roots. 5. Find the particular solution using a sinusoidal function. 6. Compute the total response of the system as the sum of the homogeneous and particular solutions. 7. Determine the constants using the initial conditions of the system being at rest. 8. Write down the final response of the system as a function of time and plot it if necessary.

Step-by-step solution

1. System representation

We can represent the mass-spring-damper system using following equation: \(m\ddot{x} + c\dot{x} + kx = F(t)\) where, - \(m\) is the mass - \(c\) is the damping coefficient - \(k\) is the spring constant - \(x\) is the displacement of the mass - \(\ddot{x}\) and \(\dot{x}\) are the acceleration and velocity of the mass respectively - \(F(t)\) is the external force

2. Force function

The force function for this problem can be defined as follows: \(F(t) = F_{0} \cdot \sin(\frac{\pi t}{t^*})\) for \(0 \le t \le t^*\) \(F(t) = F_{0}\) for \(t > t^*\)

3. Homogeneous and particular solutions

To find the response of the system, we need to find the homogeneous and particular solutions of the equation of motion: - Homogeneous solution: \(m\ddot{x_H} + c\dot{x_H} + kx_H = 0\) - Particular solution: \(m\ddot{x_P} + c\dot{x_P} + kx_P = F(t)\)

4. Homogeneous solution

The homogeneous solution can be found using the characteristic equation and assuming a solution of the form \(x_H(t) = e^{rt}\): \(r^2 + \frac{c}{m}r + \frac{k}{m} = 0\) The roots of this equation determine the nature of the homogeneous solution: - Two real and distinct roots: \(x_H(t) = C_1e^{r_1t} + C_2e^{r_2t}\) - Two complex roots: \(x_H(t) = e^{\alpha t}(C_1\cos(\beta t) + C_2\sin(\beta t))\) - Two equal real roots: \(x_H(t) = (C_1 + C_2t)e^{r_1t}\) \(C_1\) and \(C_2\) are constants to be determined using initial conditions.

5. Particular solution

For the particular solution, we can try a solution of the form: \(x_P(t) = A\sin(\omega t) + B\cos(\omega t)\) where \(\omega = \frac{\pi}{t^*}\), \(A\) and \(B\) are constants to be determined. Plug this into the equation of motion and solve for the particular solution.

6. System response

The total response of the system is the sum of the homogeneous and particular solutions: \(x(t) = x_H(t) + x_P(t)\)

7. Initial conditions

We need to find the constants \(C_1\) and \(C_2\) using the initial conditions. Let's assume the system is initially at rest, so: \(x(0) = 0\) and \(\dot{x}(0) = 0\) Plug these into the equation for the total response and solve for \(C_1\) and \(C_2\).

8. Final response

Using the results from the previous steps, write down the final response of the system as a function of time and plot it if necessary.

Key concepts

Differential Equations in Mass-Spring-Damper Systems

A mass-spring-damper system is an example of a second-order linear differential equation. This equation represents the dynamics of mechanical systems where mass, spring, and damping elements interact. The general equation can be written as: \( m\ddot{x} + c\dot{x} + kx = F(t) \). Here, \(m\) is the mass, \(c\) is the damping coefficient, \(k\) is the spring constant, and \(x\) represents the displacement.
The function \(F(t)\) is the external force applied to the system, which in this problem, transitions from a sinusoidal wave to a constant force. Understanding how to set this equation up is key to analyzing the system behavior and solving for displacement over time.

By solving this differential equation, we capture the entire system response, which includes different motion characteristics due to varying force applications.

Understanding the Homogeneous Solution

The homogeneous solution of a differential equation is where the equation equals zero (without any external force \(F(t)\)). It's essential for understanding the natural behavior of the system, characterized by \( m\ddot{x_H} + c\dot{x_H} + kx_H = 0 \).
To find this solution, start with a characteristic equation \( r^2 + \frac{c}{m}r + \frac{k}{m} = 0 \). Solving this gives you roots that describe how the system responds naturally:

• Two real and distinct roots, leading to exponential decay or growth.
• Complex roots, resulting in oscillatory behavior with damping.
• Equal roots, signifying a critically damped response.

Each root scenario impacts how the system returns to equilibrium without external influence, and this helps set the groundwork for the entire system's motion.

Finding the Particular Solution

While the homogeneous solution deals with the natural response, the particular solution considers the system’s reaction to external forces. For our system, it is given by \( m\ddot{x_P} + c\dot{x_P} + kx_P = F(t) \), where \(F(t)\) is the sinusoidal force switching to a constant one.
We assume a trial solution like \( x_P(t) = A\sin(\omega t) + B\cos(\omega t) \), solving for the amplitude and phase constants \(A\) and \(B\). These ensure the particular solution adequately reflects the specific forced motion induced by \(F(t)\).
By combining with the homogeneous solution, we derive a complete response influenced by both natural tendencies and external excitations.

Role of Initial Conditions

Initial conditions tell us the state of the system at \( t=0 \). They are critical for finding unknown constants in the solution, ensuring it accurately matches observed behavior. In this problem, we use conditions like \( x(0) = 0 \) and \( \dot{x}(0) = 0 \), typically reflecting the system starting at rest.
Applying these to both homogeneous and particular solutions helps solve for constants such as \( C_1 \) and \( C_2 \). These constants tailor the general solution to specific real-world initial scenarios, ensuring our mathematical model correctly predicts how the system will behave from its starting state.
Carefully handling initial conditions allows a predictive understanding of the system’s path based on its beginnings, bridging theory and practice effectively.