Harmonic Vibration With Viscous Damping

1.The EOM

Firstly, we should be familiar with the EOM of harmonic vibration with viscous damping, it is the basement of our discussion.
\[
m\ddot{u}(t)+c\dot{u}(t)+ku(t) = p_0sinwt
\]
The complete solution of the EOM is as follow:
\[
u(t)=e^{-\zeta w_n t}(Acosw_D t+Bsinw_D t)……①
\\ +Csinwt+Dcoswt……②
\]
where:

\[
C=\frac{P_0}{k} \frac{1-(\frac{w}{w_n})^2}{[1-(\frac{w}{w_n})^2]^2+[2\zeta\frac{w}{w_n}]^2}
\\ D = \frac{P_0}{k} \frac{-2\zeta\frac{w}{w_n}}{[1-(\frac{w}{w_n})^2]^2+[2\zeta\frac{w}{w_n}]^2}
\]

Part① is the complementary solution named Transient(which could also be regarded as Free Response ) , part② is the particular solution named Steady State. These two parts make up the Total Response of the system. (for convenience, call it EOV, which is the general solution of EOM )

The picture shows the response of damped system to harmonic force with certain parameters, it can explain why we call the complementary Transient. The Transient makes the free response decay with time, it also makes sense in math.

Before we discuss about the properties of the main parameters, I believe we have to be familiar with the DRF first.

The other form of EOM is \( m\ddot{u}(t)+c\dot{u}(t)+ku(t) = p_0coswt \) , for this Cosine form, the C , D and the general solution can be written as follows:
\[
C = \frac{P_0}{k} \frac{2\zeta\frac{w}{w_n}}{[1-(\frac{w}{w_n})^2]^2+[2\zeta\frac{w}{w_n}]^2}
\\ D = \frac{P_0}{k} \frac{1-(\frac{w}{w_n})^2}{[1-(\frac{w}{w_n})^2]^2+[2\zeta\frac{w}{w_n}]^2}
\\ u(t) = u_0 cos(wt-\phi)=(u_{st})_0 R_d cos(wt-\phi)
\]
It`s a common mathmatical transition.

2.DRF:\(R_d\),\(R_v\),\(R_a\)

2.1The relationship between three DRFs

From the EOV, we can give the following relationships by differentiating.
\[
\frac{R_a}{\frac{w}{w_n}} = R_v= \frac{w}{w_n}R_d
\]
By the way, we have already give the function \( R_d = \frac{1}{\sqrt{[1-(\frac{w}{w_n})^2}]^2+[2\zeta \frac{w}{w_n}]^2} \) as shown in the above content. We can say all the three Dynamics Response Factor are plotted as functions of \(\frac{w}{w_n}\) and \(\zeta\) (could be regarded as a constant in a system) .

Additional information: \(R_d\) is max when \(\beta^2 = (1-2\zeta^2)\) or \(w = \sqrt{1-2\zeta^2 }w_n\) , then \((R_d)_{max} = \frac{1}{2\zeta \sqrt{1-\zeta^2}}\)

2.2.The application of DRFs: Resonant Frequencies and Half-Power Bandwidth

2.2.1 Calculate the resonant frequency

Firstly we can use it to calculate the resonant frequency

Resonant frequency \(w_D\) = The forcing frequency ← The largest response amplitude occurs → means setting to 0 the first derivative of \(R_d\),\(R_v\),\(R_a\) with respect to \(\frac{w}{w_n}\) seperately.

For most occasions \(\zeta < 1/\sqrt{2}\),

Displacement resonant frequency: \(w = w_n \sqrt{1-2\zeta^2}\) → \(R_d = \frac{1}{2\zeta \sqrt{1-\zeta^2}}\)

Velocity resonant frequency: \(w = w_n\) → \(R_v = \frac{1}{2\zeta}\)

Acceleration resonant frequency: \(w = w_n /\sqrt{1-2\zeta^2}\) → \(R_a = \frac{1}{2\zeta \sqrt{1-\zeta^2}}\)

2.2.2 Evaluate the damping ratio

Secondly we can use it to evaluate the damping ratio

The half-power bandwidth is the unique property of \(R_d\) , from a complicated derivation, we can give the functions between \(w\), \(w_n\), \(\zeta\) :
\[
\zeta = \frac{w_b-w_a}{2w_n} = \frac{f_b-f_a}{2f_n}
\]
So, we can give the damping ratio without knowing the applied force, only by knowing the frequencies from forced vibration tests.

3.Properties of \(\zeta \) : The Role of Damping on the Steady State.

If we want to know the role of damping, we`d better control the variables firstly. Suppose \( w = w_0 \) and zero initial condition, we can simplified the EOV as follow:

\[
u(t)=(u_{st})_0 \frac{1}{2\zeta}[e^{-\zeta w_nt}(cosw_Dt+ \frac{\zeta}{\sqrt{1- \zeta ^2}}sinw_Dt)-cosw_nt]
\]

Still complicated, right? Never mind, as the value of \(\zeta\) is usually small in real system, the EOV still has chance for simplification. If \(\zeta → 0\) , then \(w_D ≈ w_n\) , \(\frac{\zeta}{\sqrt{1- \zeta ^2}} → 0\) ,

hence,

\[
u(t)≈ \overbrace{(u_{st})_0 \frac{1}{2\zeta}(e^{-\zeta w_nt} - 1)}^{Envolope Function}cosw_nt
\]

Therefore, the EOV would turn to convergent as \(u(t) → u_0 = (u_{st})_0 \frac{1}{2\zeta}\) , and we can also obtain \(\frac{u_j}{u_0}=1-e^{-2\pi\zeta j}\) by setting \(cosw_nt=1\) and substituting \(t=jT_n\)

So, we can draw the curves and obtain some information from them:

When \(\zeta=0.01,0.05,0.1,w=w_n.u(0)=\dot{u}(0)=0\) :

Conclusion:

①Damping lower each peak and limits the response to the bounded value;

②The lighter the damping, the larger is the number of cycles required to reach a certain percentage of \(u_o\) ,which represents the steady amplitude.

4.Properties of \(\frac{w}{w_n}\) & \(\phi\):　Maximum Deformation and Phase Lag

4.1 \(\phi\)

If we introduce a phase variable \(\phi\) and the deformation reaction factor \(R_d = \frac{u_0}{(u_{st})_0}\), we can rewrite the EOV to the most basic form of harmonic function:

\[
u(t) = u_0 sin(wt-\phi)=(u_{st})_0 R_d sin(wt-\phi)
\]
This form can easily tell us: \(u_0 = \sqrt{C^2+D^2} \) , \( \phi = tan^{-1}(-\frac{D}{C})=tan^{-1} \frac{2\zeta\frac{w}{w_n}}{1-(\frac{w}{w_n})^2} \) , \(R_d = \frac{1}{\sqrt{[1-(\frac{w}{w_n})^2}]^2+[2\zeta \frac{w}{w_n}]^2} \)

If we plot the equation of \(R_d\) in the following figs for three values of \(\frac{w}{w_n}\) and a fixed value of \(\zeta = 0.20\) , we can find that:

The steady state motion is seen to occer at the forcing period \(T = \frac{2\pi}{w}\) , but with a time lag \(= \frac{\phi}{2\pi}\)

4.2 \(\frac{w}{w_n}\)

According to the frequency-response curve, let us examine the three regions of the excitation- frequency scale:

When \(\frac{w}{w_n} \ll 1\) , means the force is slowly varying:

\(R_d\) is only slightly larger than 1 and is essentially independent of damping;
\[
u_o ≈ (u_{st})_0 = \frac{p_o}{k}
\]
The amplitude of dynamic response is the same as the static deformation and is controlled by the stiffness of the system.

\(\phi\) is close to 0° and the displacement is in phase with the applied force.

When \(\frac{w}{w_n} \gg 1\) , means the force is rapidly varyng:

\(R_d\) tends to zero and is unaffected by damping.
\[
u_o ≈ (u_{st})_0 \frac{w_n^2}{w^2} = \frac{p_o}{mw^2}
\]
The response is controlled by the mass of the system.

\(\phi\) is close to 180° and the displacement is out of phase with the applied force.

When \( \frac{w}{w_n} ≈ 1 \) :

\(R_d\) is sensitive to damping;

For the smaller damping value, \(R_d\) can be several times larger than 1 and the amplitude of dynamic response can be much larger than the static deformation.
\[
u_o ≈ (u_{st})_0 \frac{1}{2\zeta} = \frac{p_o}{cw_n}
\]
The response is controlled by the damping of the system.

\(\phi\) is close to 90° and the displacement attains its peaks when the force passes through zeros.

The conclusions above are hard to remember directly, we`d better remember the following figures: