Mass-Spring-Damper System

Problem Statement

We have a damped mass-spring system with a natural frequency of 1 rad/s and damping ratio of 0.1. An initial velocity of -1 cm/s is given to the mass. What is the minimum displacement that occurs during the motion?

Mathematical Model

The displacement of a mass-spring-damper system with initial displacement zero and initial velocity $v_0$ is given by:

$x(t)=e^{-\xi {\omega }_{n}t}\frac{v_0}{{\omega }_d}\sin ({\omega }_{d}t)$

Where:

• $\xi$ is the damping ratio (0.1 in this problem)
• ${\omega }_n$ is the natural frequency (1 rad/s in this problem)
• $t$ is the time
• $v_0$ is the initial velocity (-1 cm/s in this problem)
• ${\omega }_d={\omega }_n\sqrt{1-{\xi }^2}$ is the damped natural frequency

Optimisation Setup

Design Variable

• $t$ = time at which minimum displacement occurs (scalar, 1-dimensional problem)

Objective Function

We want to find the minimum (most negative) displacement, which means minimizing:

$L(t):=x(t)=e^{-\xi {\omega }_{n}t}\frac{v_0}{{\omega }_d}\sin ({\omega }_{d}t)$

Since $v_0$ is negative (-1 cm/s), we’re looking for the time when $\sin ({\omega }_{d}t)$ is most positive to give the most negative displacement.

Solution Approach

To find the time at which the minimum displacement occurs, we differentiate $x(t)$ with respect to $t$ and set it to zero:

$\frac{dx}{dt}=e^{-\xi {\omega }_{n}t}\frac{v_0}{{\omega }_d}{\omega }_d\cos ({\omega }_{d}t)-\xi {\omega }_n{e}^{-\xi {\omega }_{n}t}\frac{v_0}{{\omega }_d}\sin ({\omega }_{d}t)=0$

Simplifying:

$\frac{v_0}{{\omega }_d}{e}^{-\xi {\omega }_{n}t}[{\omega }_d\cos ({\omega }_{d}t)-\xi {\omega }_n\sin ({\omega }_{d}t)]=0$

Since $e^{-\xi {\omega }_{n}t}$ is never zero and $v_0/{\omega }_d$ is a constant, we need:

${\omega }_d\cos ({\omega }_{d}t)-\xi {\omega }_n\sin ({\omega }_{d}t)=0$

This can be rearranged to:

$\frac{\cos ({\omega }_{d}t)}{\sin ({\omega }_{d}t)}=\frac{\xi {\omega }_n}{{\omega }_d}$

$\tan ({\omega }_{d}t)=\frac{{\omega }_d}{\xi {\omega }_n}$

Solving for $t$:

$t^{\ast }=\frac{1}{{\omega }_d}{\tan }^{-1}\left(\frac{{\omega }_d}{\xi {\omega }_n}\right)+\frac{n\pi }{{\omega }_d}$

where $n$ is an integer. For the first minimum, $n=0$.

Numerical Solution for Our Problem

Given parameters:

• ${\omega }_n=1$ rad/s
• $\xi =0.1$
• $v_0=-1$ cm/s

First, we calculate the damped natural frequency: ${\omega }_d={\omega }_n\sqrt{1-{\xi }^2}=1\times \sqrt{1-0.1^2}=0.995\text{ rad/s}$

Next, we find the time of minimum displacement: $t^{\ast }=\frac{1}{0.995}{\tan }^{-1}\left(\frac{0.995}{0.1\times 1}\right)=\frac{1}{0.995}{\tan }^{-1}(9.95)\approx 1.57\text{ s}$

The minimum displacement is: $x(t^{\ast })=e^{-0.1\times 1\times 1.57}\frac{-1}{0.995}\sin (0.995\times 1.57)\approx -0.90\text{ cm}$

Physical Interpretation

1. The mass-spring-damper oscillates with decaying amplitude due to damping
2. The time of first minimum displacement is approximately π/(2ωd) seconds after the start
3. After this point, the displacement still oscillates but with decreasing amplitude
4. Eventually, the system comes to rest at the equilibrium position (x = 0)

Second Derivative Test

To verify this is indeed the minimum displacement (most negative value), we check the second derivative at $t^{\ast }$:

$\frac{d^{2}x}{d{t}^2}{|}_{t=t^{\ast }}>0$

If this condition is met, then $t^{\ast }$ gives us the minimum displacement.

Generalization

For a mass-spring-damper system with arbitrary parameters and initial conditions, the times of extrema can be found by solving:

$\tan ({\omega }_{d}t)=\frac{{\omega }_d}{\xi {\omega }_n\pm \frac{{\omega }_d{x}_0}{v_0}}$

where the sign depends on whether we’re looking for minima or maxima, and $x_0$ is the initial displacement.

Key Insights

1. This problem demonstrates the application of optimization to dynamic systems
2. The objective function is time-dependent, making this a different type of optimization problem
3. The solution requires both calculus and understanding of dynamic system behavior
4. The damped oscillator represents many real engineering systems (vehicle suspensions, building structures, etc.)
5. The optimization gives insight into the “worst-case” displacement that the system will experience