User:Lzyvzl/Vibration Draft

Vibration refers to mechanical oscillations about an equilibrium point. The oscillations may be periodic such as the motion of a pendulum or random such as the movement of a tire on a gravel road.

Vibration is occasionally desirable. For example the motion of a tuning fork, the reed in a woodwind instrument or harmonica, or the cone of a loudspeaker is desirable vibration, necessary for the correct functioning of the various devices.

More often, vibration is undesirable, wasting energy and creating unwanted sound--noise. For example, the motions of engines, electric motors, or any mechanical device in operation are usually unwanted vibrations. Such vibrations can be caused by imbalances in the rotating parts, uneven friction, the meshing of gear teeth, parts that are dragging together, etc. Careful designs usually minimize unwanted vibrations.

The study of sound and vibration are closely related. Sound, pressure waves, are generated by vibrating structures (e.g. speaker cone) and pressure waves can generate vibration of structures (e.g. ear drum). Hence, when trying to reduce noise it is often a problem in trying to reduce vibration.

Types of vibration

Free vibration occurs when a mechanical system is set off with an initial input and then allowed to vibrate freely. Examples of this type of vibration are pulling a child back on a swing and then letting go or hitting a tuning fork and letting it ring. The mechanical system will then vibrate at one or more of its natural frequencies and damp down to zero.

Forced vibration is when an alternating force or motion is applied to a mechanical system. Examples of this type of vibration are a shaking washing machining due to an imbalance or the vibration of a building during an earthquake. In forced vibration the frequency of the vibration is dependent on the frequency content of the force or motion applied, but the magnitude of the vibration is strongly dependent on the behavior of the mechanical system.

Vibration analysis

The foundation of vibration analysis is the study of the simple mass-spring-damper model. Indeed, even a complex structure such as an automobile body can be modeled as a summation of simple mass-spring-damper models. The mass-spring-damper model is an example of a simple harmonic oscillator and hence the mathematics used to describe its behavior is identical to other simple harmonic oscillators such as the RLC electric circuit.

File:Mass spring damper.png

Simple Mass Spring Damper Model

An ideal mass-spring-damper system with mass m, spring stiffness k and damping coefficient c.

To start we will assume the damping is negligilbe and focus only on the mass and spring.

The force generated by the spring is proportional to the displacement "x".

F_{\mathrm {s} }\ \ =\ \ -kx

The force generated by mass is proportional to the acceleration. : $\Sigma \ F\ \ =\ \ ma\ \ =\ \ m{\ddot {x}}\ \ =\ \ m{\frac {d^{2}x}{dt^{2}}}$

where a is the acceleration of the mass and $x$ is the displacement of the mass relative to a fixed point of reference.

F_{\mathrm {d} }\ \ =\ \ -cv\ \ =\ \ -c{\dot {x}}\ \ =\ \ -c{\frac {dx}{dt}}

Differential equation

The equations of motion combine to form a second-order differential equation for displacement x as a function of time t (in seconds):

m{\ddot {x}}+c{\dot {x}}+kx=0

Rearranging, we have

{\ddot {x}}+{c \over m}{\dot {x}}+{k \over m}x=0.

Next, to simplify the equation, we define the following parameters:

\omega _{0}={\sqrt {k \over m}}

and

\zeta ={c \over 2{\sqrt {km}}}.

The first parameter, ω₀, is called the (undamped) natural frequency of the system. The second, ζ, is called the damping factor. The natural frequency represents an angular frequency and has for units of measure radians per second. The damping factor is a dimensionless quantity.

The differential equation now becomes

{\ddot {x}}+2\zeta \omega _{0}{\dot {x}}+\omega _{0}^{2}x=0.

Continuing, we can solve the equation by assuming

\ x=e^{\gamma t}\

where the parameter $\ \gamma \$ is, in general, a complex number.

Substituting this assumed solution back into the differential equation, we obtain

\gamma ^{2}+2\zeta \omega _{0}\gamma +\omega _{0}^{2}=0.

Solving for γ, we find:

\gamma =\omega _{0}(-\zeta \pm {\sqrt {\zeta ^{2}-1}}).

System behavior

The behavior of the system depends on the relative values of the two fundamental parameters, the natural frequency ω₀ and the damping factor ζ. In particular, the qualitative behavior of the system depends crucially on whether the quadratic equation for $\gamma$ has one real solution, two real solutions, or two complex conjugate solutions.

Critical damping

When $\zeta =1$ , $\gamma \$ (defined above) is real and the system is critically damped. An example of critical damping is the door-closer seen on many hinged doors in public buildings.

Over-damping

When $\zeta >1$ , $\gamma \$ is still real, but now the system is said to be over-damped. An overdamped door-closer will take longer to close the door than a critically damped door closer.

Under-damping

Finally, when $\zeta <1,$ $\gamma \$ is complex, and the system is under-damped. In this situation, the system will oscillate at the damped frequency $\omega _{\mathrm {d} }=\omega _{0}{\sqrt {1-\zeta ^{2}}}$ , which is a function of the natural frequency and the damping factor.

Solution

In the underdamped case, the solution can be generally written as:

x(t)\ =\ Ae^{-\zeta \omega _{0}t}\cos(\omega _{\mathrm {d} }t+\varphi )

where

\omega _{\mathrm {d} }=\omega _{0}{\sqrt {1-\zeta ^{2}}}

represents the damped frequency of the system, and A and φ are determined by the initial conditions of the system (usually the initial position and velocity of the mass).

In the critically damped case, the solution takes the form

x(t)\ =\ (A+Bt)e^{-\zeta \omega _{0}t}

where A and B are again determined by the initial conditions.