### Vibration Damping

Damping is an influence within or upon an oscillatory system that has the effect of reducing, restricting or preventing its oscillations. In physical systems, damping is produced by processes that dissipate the energy stored in the oscillation. Examples include Viscous drag in mechanical systems, resistance in electronic oscillators, and absorption and scattering of light in optical oscillators. Damping not based on energy loss can be important in other oscillating systems such as those that occur in biological systems.

The damping of a system can be described as being one of the following:

• Overdamped: The system returns (exponentially decays) to equilibrium without oscillating.
• Critically damped: The system returns to equilibrium as quickly as possible without oscillating.
• Underdamped: The system oscillates (at reduced frequency compared to the undamped case) with the amplitude gradually decreasing to zero.
• Undamped The system oscillates at its natural resonant frequency (ωo).

For example, consider a door that uses a spring to close the door once open. This can lead to any of the above types of damping depending on the strength of the damping. If the door is undamped it will swing back and forth forever at a particular resonant frequency. If it is underdamped it will swing back and forth with decreasing size of the swing until it comes to a stop. If it is critically damped then it will return to closed as quickly as possible without oscillating. Finally, if it is overdamped it will return to closed without oscillating but slower depending on how overdamped it is. Different levels of damping are desired for different types of systems.

## Linear Damping

A particularly mathematically useful type of damping is linear damping. Linear damping occurs when a potentially oscillatory variable is damped by an influence that opposes changes in it, in direct proportion to the instantaneous rate of change, velocity or time derivative, of the variable itself. In engineering applications it is often desirable to linearize non-linear drag forces. This may be done by finding an equivalent work coefficient in the case of harmonic forcing. In non-harmonic cases, restrictions on the speed may lead to accurate linearization.

In physics and engineering, damping may be mathematically modelled as a force synchronous with the velocity of the object but opposite in direction to it. If such force is also proportional to the velocity, as for a simple mechanical viscous damper (dashpot), the force $F$ may be related to the velocity $v$ by

$F = -cv \, ,$

where c is the damping coefficient, given in units of newton-seconds per meter.

This force may be used as an approximation to the friction caused by drag and may be realized, for instance, using a dashpot. (This device uses the viscous drag of a fluid, such as oil, to provide a resistance that is related linearly to velocity.) Even when friction is related to $v^2$, if the velocity is restricted to a small range, then this non-linear effect may be small. In such a situation, a linearized friction coefficient $c_\left\{lin\right\}$ may be determined which produces little error.

When including a restoring force (such as due to a spring) that is proportional to the displacement $x$ and in the opposite direction, and by setting the sum of these two forces equal to the mass of the object times its acceleration creates a second-order differential equation whose terms can be rearranged into the following form:

$\frac\left\{d^2x\right\}\left\{dt^2\right\} + 2\zeta\omega_0\frac\left\{dx\right\}\left\{dt\right\} + \omega_0^2 x = 0,$

where ω0 is the undamped angular frequency of the oscillator and ζ is a constant called the damping ratio. This equation is valid for many different oscillating systems, but with different formulas for the damping ration and the undamped angular frequency.

The value of the damping ratio ζ determines the behavior of the system such that ζ = 1 corresponds to being critically damped with larger values being overdamped and smaller values being underdamped. If ζ = 0, the system is undamped.

### Example: mass–spring–damper

An ideal mass–spring–damper system with mass m, spring constant k and viscous damper of damping coefficient c is subject to an oscillatory force

$F_\mathrm\left\{s\right\} = - k x \,$

and a damping force

$F_\mathrm\left\{d\right\} = - c v = - c \frac\left\{dx\right\}\left\{dt\right\} = - c \dot\left\{x\right\}.$

The values can be in any consistent system of units; for example, in SI units, m in kilograms, k in newtons per meter, and c in newton-seconds per meter or kilograms per second.

Treating the mass as a free body and applying Newton's second law, the total force Ftot on the body is

$F_\mathrm\left\{tot\right\} = ma = m \frac\left\{d^2x\right\}\left\{dt^2\right\} = m \ddot\left\{x\right\}.$

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

Since Ftot = Fs + Fd,

$m \ddot\left\{x\right\} = -kx + -c\dot\left\{x\right\}.$

This differential equation may be rearranged into

$\ddot\left\{x\right\} + \left\{ c \over m\right\} \dot\left\{x\right\} + \left\{k \over m\right\} x = 0.\,$

The following parameters are then defined:

$\omega_0 = \sqrt\left\{ k \over m \right\}$
$\zeta = \left\{ c \over 2 \sqrt\left\{m k\right\} \right\}.$

The first parameter, ω0, is called the (undamped) natural frequency of the system . The second parameter, ζ, is called the damping ratio. The natural frequency represents an angular frequency, expressed in radians per second. The damping ratio is a dimensionless quantity.

The differential equation now becomes

$\ddot\left\{x\right\} + 2 \zeta \omega_0 \dot\left\{x\right\} + \omega_0^2 x = 0.\,$

Continuing, we can solve the equation by assuming a solution x such that:

$x = e^\left\{\gamma t\right\}\,$

where the parameter γ (gamma) is, in general, a complex number.

Substituting this assumed solution back into the differential equation gives

$\gamma^2 + 2 \zeta \omega_0 \gamma + \omega_0^2 = 0 \, ,$

which is the characteristic equation.

Solving the characteristic equation will give two roots, γ+ and γ. The solution to the differential equation is thus



x(t) = Ae^{\gamma_+ t} + Be^{\gamma_- t} \, ,

where A and B are determined by the initial conditions of the system:



A = x(0)+\frac{\gamma_+x(0)-\dot{x}(0)}{\gamma_--\gamma_+}



B = -\frac{\gamma_+x(0)-\dot{x}(0)}{\gamma_--\gamma_+}.

### System behaviour

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

#### Critical damping (ζ = 1)

When ζ = 1, there is a double root γ (defined above), which is real. The system is said to be critically damped. A critically damped system converges to zero as fast as possible without oscillating. An example of critical damping is the door closer seen on many hinged doors in public buildings. The recoil mechanisms in most guns are also critically damped so that they return to their original position, after the recoil due to firing, in the least possible time.

In this case, with only one root γ, there is in addition to the solution x(t) = eγt a solution x(t) = teγt:



x(t) = (A+Bt)\,e^{-\omega_0 t} \,

where $A$ and $B$ are determined by the initial conditions of the system (usually the initial position and velocity of the mass):



A = x(0) \,



B = \dot{x}(0)+\omega_0x(0) \,

#### Over-damping (ζ > 1)

When ζ > 1, the system is over-damped and there are two different real roots. An over-damped door-closer will take longer to close than a critically damped door would.

The solution to the motion equation is:



x(t) = Ae^{\gamma_+ t} + Be^{\gamma_- t}

where $A$ and $B$ are determined by the initial conditions of the system:



A = x(0)+\frac{\gamma_+x(0)-\dot{x}(0)}{\gamma_--\gamma_+}



B = -\frac{\gamma_+x(0)-\dot{x}(0)}{\gamma_--\gamma_+}.

#### Under-damping (0 ≤ ζ < 1)

Finally, when 0 < ζ < 1, γ is complex, and the system is under-damped. In this situation, the system will oscillate at the natural damped frequency ωd, which is a function of the natural frequency and the damping ratio. To continue the analogy, an underdamped door closer would close quickly, but would hit the door frame with significant velocity, or would oscillate in the case of a swinging door.

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

$x \left(t\right) = e^\left\{- \zeta \omega_0 t\right\} \left(A \cos\,\left(\omega_\mathrm\left\{d\right\}\,t\right) + B \sin\,\left(\omega_\mathrm\left\{d\right\}\,t \right)\right)\,$

where

$\omega_\mathrm\left\{d\right\} = \omega_0 \sqrt\left\{1 - \zeta^2 \right\}\,$

represents the damped frequency or ringing frequency of the system, and A and B are again determined by the initial conditions of the system:

$A = x\left(0\right)\,$
$B = \frac\left\{1\right\}\left\{\omega_\mathrm\left\{d\right\}\right\}\left(\zeta\omega_0x\left(0\right)+\dot\left\{x\right\}\left(0\right)\right).\,$

This "damped frequency" is not to be confused with the damped resonant frequency or peak frequency ωpeak. This is the frequency at which a moderately underdamped (ζ < 1/2) simple 2nd-order harmonic oscillator has its maximum gain (or peak transmissibility) when driven by a sinusoidal input. The frequency at which this peak occurs is given by:

$\omega_\left\{peak\right\} = \omega_0\sqrt\left\{1 - 2\zeta^2\right\}$.

For an under-damped system, the value of ζ can be found by examining the logarithm of the ratio of succeeding amplitudes of a system. This is called the logarithmic decrement.

## Alternative models

Viscous damping models, although widely used, are not the only damping models. A wide range of models can be found in specialized literature. One is the so-called "hysteretic damping model" or "structural damping model".

When a metal beam is vibrating, the internal damping can be better described by a force proportional to the displacement but in phase with the velocity. In such case, the differential equation that describes the free movement of a single-degree-of-freedom system becomes:



m \ddot{x} + h x \imath + k x = 0

where h is the hysteretic damping coefficient and i denotes the imaginary unit; the presence of i is required to synchronize the damping force to the velocity (xi being in phase with the velocity). This equation is more often written as:



m \ddot{x} + k ( 1 + \imath \eta ) x = 0

where η is the hysteretic damping ratio, that is, the fraction of energy lost in each cycle of the vibration.

Although requiring complex analysis to solve the equation, this model reproduces the real behaviour of many vibrating structures more closely than the viscous model.

A more general model that also requires complex analysis, the fractional model not only includes both the viscous and hysteretic models, but also allows for intermediate cases (useful for some polymers):



m \ddot{x} + A \frac{d^r x}{dt^r} \imath + k x = 0

where r is any number, usually between 0 (for hysteretic) and 1 (for viscous), and A is a general damping (h for hysteretic and c for viscous) coefficient.

### Nonlinear damping

Nonlinear passive damping offers important advantages compared to purely linear designs. Nonlinear damping using an odd function, for example cubic damping, allows the user to damp the resonance without increasing the energy in the frequency response tails and hence overcomes several limitations of a purely linear design.