Viscous Damping
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## 1 Introduction to Viscous Damping[1]

Figure 1: Single DOF model with Viscous Damping

Viscous damping is a common form of damping which is formed in many engineering systems such as instruments adn shock absorbers. The viscous damping force is proportional to the first power of the velocity across the damper, and it always opposes the motion, so that the damping force is a linear continuous function of the velocity. Because the analysis of viscous damping leads to the simplest mathematical treatment, analysts sometimes approxiate more complex types of damping to the viscous type.

Consider the single degree of freedom model with viscous damping shown in Figure 1. The only unfamiliar element in the system is the viscous damper with coefficient c. This coefficient is such that the damping force required to move the body with a velocity $LaTeX: \dot{x}$ is $LaTeX: c\dot{x}$.

Figure 2: Single DOF model with Viscous Damping, Free Body Diagram

For motion of the body in the direction shown, the free body diagrams are as in Figure 2. The equation of motion is therefore

 $LaTeX: m\ddot{x}+c\dot{x}+kx=0$ (2.9)

This equation of motion pertains to the whole of the cycle: the reader should verify that this is so. (Note: displacements to the left of the equilibrium position are negative and velocities and accelerations from right to left are also negative.) Equation (2.9) is a 2nd order differential equation which can be solved by assuming a solution of the form $LaTeX: x=Xe^{st}$. Substituting this solution into equation (2.9) gives

 $LaTeX: \left(ms^2+cs+k\right)Xe^{st}=0$

Since $LaTeX: Xe^{st} ne 0$ (otherwise no motion)

 $LaTeX: ms^2+cs+k=0$

so,

 $LaTeX: s_{1, 2}=-\frac{c}{2m} \pm \frac{\sqrt{c^2 - 2mk}}{2m}$

Hence

 $LaTeX: x=X_{1}e^{s_1t}+X_{2}e^{s_2t}$

Where X1 and X2 are arbitrary constants found from the initial conditions. The system response evidently depends on whether c is positive or negative, and on whether c2 is greater than, equal to, or less than 4mk.

The dynamic behavior of the system depends on the numerical value of the radical, so we define critical damping as that value of c(cc) which makes the radical zero: that is,

 $LaTeX: c_c=2\sqrt{km}$

Hence

 $LaTeX: \frac{c_c}{2m}=2\sqrt{\frac{k}{m}}=\omega_n$ Undamped Natural Frequency

 $LaTeX: c_c=2\sqrt{km}=2m\omega_n$

The actual damping is a system can be specified in terms of cc by introducing the damping ratio ζ. Thus

 $LaTeX: \zeta=\frac{c}{c_c}$

and

 $LaTeX: s_{1, 2}=\left [-\zeta \pm \sqrt{\left(\zeta^2-1\right)}\right ] \omega_n$ (2.10)

The response depends on whether or not c is positive or negative, and on whether ζ is grater than to, or less than 1. Usually c is positive, so we need consider only the other possibilities.

### 1.1 Viscous Damping Case 1: $LaTeX: \zeta<1$

Figure 3: Bode magnitude of ideal 2nd order Transfer Function at different damping ratios
Figure 4: Bode phase of ideal 2nd order Transfer Function at different damping ratios

When $LaTeX: \zeta<1$ the damping is less than critical. From Eqn. (2.10)

 $LaTeX: s_{1, 2}=-\zeta\omega_n \pm j\sqrt{\left(1-\zeta^2\right)}\omega_n$

so

 $LaTeX: x=e^{-\zeta\omega_n t}\left [ X_1e^{j\sqrt{1-\zeta^2}\omega_n t}+X_2e^{-j\sqrt{1-\zeta^2}\omega_n t}\right ]$

and

 $LaTeX: x=Xe^{-\zeta\omega_n t}\mbox{sin}\left(\sqrt{1-\zeta^2}\omega_n t+\phi\right)$

The motion of the body is therefore an exponentionally decaying harmonic oscillation with circular frequency $LaTeX: \omega_v=\omega_n\sqrt{1-\zeta^2}$.

The frequency of the viscously damped oscillation $LaTeX: \omega_v$, is given by $LaTeX: \omega_v=\omega_n\sqrt{1-\zeta^2}$, that is, the frequency of oscillation is reduced by the damping action. However, in many systems this reduction is likely to be small, because very small values of ζ are common; for example in most engineering structures ζ is rarely greater than 0.02. Even if $LaTeX: \zeta=0.2$, $LaTeX: \omega_v=0.98\omega_n$. This is not true in those cases where ζ is large, for example in motor vehicles where ζ is typically 0.7 for new shock absorbers.

### 1.2 Viscous Damping Case 2: $LaTeX: \zeta=1$

Viscous damping Case 2 is critically damped. Both values of s are - ω. However, two constants are required in the solution of equation (2.9), thus $LaTeX: x=\left(A+Bt\right)e^{-\omega_n t}$ may be assumed.

Critical damping represents the limit of periodic motion, hence the displaced body is restored to equilibrium in the shortest possible time \, and without oscilllation or overshoot. Many devices, particularly electrical instruments, are critically damped to take advantage of this property.

### 1.3 Viscous Damping Case 3: $LaTeX: \zeta>1$

Viscous damping Case 3 is over damped. There are two real values of s, so $LaTeX: x=X_1e^{s_1t}+X_2e^{s_2t}$. Since both values of s are negative the motion is the sum of two exponential decays.