What do mechanical vibrations, oscillations, force, mass, damping, and resonance have in common?
Jenn, Founder Calcworkshop®, 15+ Years Experience (Licensed & Certified Teacher)
They are all derived with the use of differential equations and Newton’s law of motion.
Simple Harmonic Motion and Equilibrium
A mechanical vibration is defined as the measurement of a periodic process of oscillations with respect to an equilibrium point.
But what does this really mean?
Simple harmonic motion, sometimes called free undamped motion, occurs naturally.
Why?
Because many objects have a natural vibratory motion that oscillates back and forth about a fixed point. Whenever this happens, the oscillating object is met with negligible resistance or damping.
A pendulum is an example of such an occurrence, and the center line in which the pendulum or object moves back and forth is called the equilibrium position of the object’s motion, whereas the farthest displacement of the object from the equilibrium position is called the amplitude.
Different Types of Motion
But there is more than just simple harmonic motion, as there is angular motion, projectile motion, and helical (spring) motion too.
In this lesson, we will be looking at various types of motion all while explaining definition, notations, and formulas, as they pertain to our study of differential equations.
In particular, we will learn that a damped harmonic motion dissipates over time and see how this effects frequency and resonance.
Why?
Transient and Steady State Motion
Well, motion due to the complementary (homogeneous) function, \(y_{h}\), whether oscillatory or non-oscillatory, dies out over time. That is why \(y_{h}\) is sometimes called the transient motion.
Which means is that motion is only tricky to calculate during the time in which the transient motion is active or effective. Afterward, the motion is determined entirely by the particular solution \(y_{p}\). Thus, the particular solution is sometimes called the steady state motion.
Critical Damping, Underdamped, and Overdamped Systems
Additionally, we will discover that critical damping is when we return a system to equilibrium as fast as possible, whereas underdamped systems will oscillate through the equilibrium position and overdamped systems will move slowly toward equilibrium.
Some of the formulas and properties that we will be encountering are as follows:
Hooke’s Law
Differential Equation:
\[\begin{equation}
m \frac{d^{2} y}{d t^{2}}=-k y \text{ or } \frac{d^{2} y}{d t^{2}}+\frac{k}{m} y=0
\end{equation}\]
Solutions:
\[\begin{equation}
y=c \cos \left(t \sqrt{\frac{k}{m}}+\delta\right) \text{ or } y=c \sin \left(t \sqrt{\frac{k}{m}}+\delta\right)
\end{equation}\]
Simple Harmonic or Free Undamped Motion
Differential Equation:
\[\begin{equation}
\frac{d^{2} y}{d t^{2}}+\omega_{0}^{2} y=0
\end{equation}\]
Solutions:
\[\begin{equation}
y_{h}=c_{1} \cos \left(\omega_{0} t\right)+c_{2} \sin \left(\omega_{0} t\right) \text{ or } y_{h}=c \cos \left(\omega_{0} t+\delta\right) \text{ or } y_{h}=c \sin \left(\omega_{0} t+\delta\right)
\end{equation}\]
Stable perpetual motion with Amplitude:
\[\begin{equation}
A=\sqrt{c_{1}^{2}+c_{2}^{2}}
\end{equation}\]
Frequency:
\[\begin{equation}
\frac{\omega_{0}}{2 \pi} \text{ cycles per unit of time}
\end{equation}\]
Forced Undamped Motion
Differential Equation:
\[\begin{equation}
\frac{d^{2} y}{d t^{2}}+\omega_{0}^{2} y=F \sin (\omega t+\beta)
\end{equation}\]
Solutions:
\[\begin{align}
y &= c_{1} \cos \left(\omega_{0} t\right)+c_{2} \sin \left(\omega_{0} t\right)+\frac{F}{\omega_{0}^{2}-\omega^{2}} \sin (\omega t+\beta), \quad \omega_{0} \neq \omega \\
y &= c_{1} \cos \left(\omega_{0} t\right)+c_{2} \sin \left(\omega_{0} t\right)-\frac{F}{2 \omega_{0}} t \cos (\omega t+\beta), \quad \omega_{0}=\omega
\end{align}\]
Stable, Oscillatory motion with Amplitude:
\[\begin{equation}
A=c+\frac{F}{\omega_{0}^{2}-\omega^{2}}
\end{equation}\]
Unstable (Resonance) motion with Amplitude:
\[\begin{equation}
A=c-\frac{F}{2 \omega_{0}} t
\end{equation}\]
Damped Harmonic Motion or Free Damped Motion
Differential Equation:
\[\begin{equation}
\frac{d^{2} y}{d t^{2}}+2 r \frac{d r}{d t}+\omega_{0}^{2}=0
\end{equation}\]
Solutions:
\[\begin{align*}
\begin{aligned}
& y_{h}=c_{1} e^{\left(-r+\sqrt{r^{2}-\omega_{0}^{2}}\right)}+c_{2} e^{\left(-r-\sqrt{r^{2}-\omega_{0}^{2}}\right)}, \quad r>\omega_{0} \\
& y_{h}=c_{1} e^{-r t}+c_{2} t e^{-r t}, \quad r=\omega_{0}
\end{aligned}
\end{align*}\]
\[\begin{equation}
y_{h}=c e^{-r t} \sin \left(\delta+t \sqrt{\omega_{0}^{2}-r^{2}}\right), \quad r<\omega_{0}
\end{equation}\]
Stable, Oscillatory motion with Amplitude:
\[\begin{equation}
A=c e^{-r t}
\end{equation}\]
Frequency:
\[\begin{equation}
\frac{\sqrt{\omega_{0}^{2}-r^{2}}}{2 \pi} \text{ cycles per unit of time}
\end{equation}\]
Forced Damped Motion
Differential Equation:
\[\begin{equation}
\frac{d^{2} y}{d t^{2}}+2 r \frac{d r}{d t}+\omega_{0}^{2} y=F \sin (\omega t+\beta)
\end{equation}\]
Solutions:
\[\begin{equation}
y=c_{1} \cos \left(\omega_{0} t\right)+c_{2} \sin \left(\omega_{0} t\right)+\frac{F \sin (\omega t+\beta-\alpha)}{\sqrt{\left(\omega_{0}^{2}-\omega^{2}\right)^{2}+(2 r \omega)^{2}}}
\end{equation}\]
Stable, Oscillatory motion with Amplitude:
\[\begin{equation}
A=c+\frac{F}{\sqrt{\left(\omega_{0}^{2}-\omega^{2}\right)^{2}+(2 r \omega)^{2}}}
\end{equation}\]
Steady state motion with Frequency:
\[\begin{equation}
\frac{\omega}{2 \pi} \text{ cycles per unit of time}
\end{equation}\]
Lesson Overview
Great! Together, we’ll cover the following topics in our lesson:
- Review Hooke’s Law for a mass attached to a spring
- Understand equilibrium positions
- Understand positive and negative orientation
- Derive the differential equation using Hooke’s Law and Newton’s Second Law of Motion
- Find a second-order, linear, differential equation
- Determine the displacement of an object
- Compare free vs. forced and damped vs. undamped motion
- Comprehend critical damping
- Understand over damping and under damping
- Solve Forced-Undamped and Forced-Damped Vibration problems
- Learn a handy trick for finding the Particular Solution
- Avoid relying on Undetermined Coefficients or Variations of Parameters
There’s a lot to cover, so let’s dive right in!
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