Transmissibility in Vibrating Systems

The concept of Transmissibility is crucial in the study of mechanical vibrations, especially when designing systems for vibration isolation. It represents the ratio of the amplitude of the transmitted force (or motion) to the amplitude of the exciting force (or motion). In simpler terms, it tells us how much of the vibratory input gets "transmitted" through the system. A low transmissibility value indicates effective vibration isolation, meaning less vibration is passed on.

For a forced vibration system with viscous damping, the transmissibility (TR), also known as the amplification factor, is a function of the frequency ratio and the damping ratio.

Understanding the Transmissibility Formula

The general formula for transmissibility ($TR$) in a single-degree-of-freedom system subjected to harmonic excitation is given by:

$$ TR = \frac{1}{\sqrt{\left[1 - \left(\frac{\omega}{\omega_n}\right)^2\right]^2 + \left[2\zeta\left(\frac{\omega}{\omega_n}\right)\right]^2}} $$

Where:

• \(\omega\) is the frequency of the exciting force.
• \(\omega_n\) is the natural frequency of the vibrating system.
• \(\zeta\) (zeta) is the damping ratio of the system.
• The ratio \(\frac{\omega}{\omega_n}\) is known as the frequency ratio, often denoted by \(r\).

In the special case of an undamped system (\(\zeta = 0\)), the formula simplifies significantly to:

$$ TR = \frac{1}{\left|1 - \left(\frac{\omega}{\omega_n}\right)^2\right|} $$

Analyzing the Frequency Ratio for Transmissibility

The question asks about the transmissibility when the frequency ratio \(\left(\frac{\omega}{\omega_n}\right)\) is greater than \(\sqrt{2}\). Let's denote the frequency ratio as \(r = \frac{\omega}{\omega_n}\). So, the condition is \(r > \sqrt{2}\).

If \(r > \sqrt{2}\), then squaring both sides gives us \(r^2 > (\sqrt{2})^2\), which means \(r^2 > 2\).

Calculating Transmissibility Beyond Resonance

Case 1: Undamped System (\(\zeta = 0\))

For an undamped system, we use the simplified formula:

$$ TR = \frac{1}{\left|1 - r^2\right|} $$

Since we established that \(r^2 > 2\), the term \(1 - r^2\) will be a negative number (e.g., if \(r^2 = 3\), then \(1 - r^2 = 1 - 3 = -2\)).

Therefore, the absolute value \(\left|1 - r^2\right|\) becomes \( -(1 - r^2) \), which simplifies to \( r^2 - 1 \).

So, for the undamped case when \(r > \sqrt{2}\):

$$ TR = \frac{1}{r^2 - 1} $$

Now, let's analyze the denominator \(r^2 - 1\):

• Since \(r^2 > 2\), it implies that \(r^2 - 1 > 2 - 1\).
• This means \(r^2 - 1 > 1\).

If the denominator \(r^2 - 1\) is greater than 1, then the fraction \(\frac{1}{r^2 - 1}\) must be less than one.

For example, if \(r = 2\), then \(TR = \frac{1}{2^2 - 1} = \frac{1}{4 - 1} = \frac{1}{3}\). Since \(\frac{1}{3} < 1\), this confirms that the transmissibility is less than one.

Case 2: Damped System (\(\zeta > 0\))

Now, let's consider the general formula for a system with damping:

$$ TR = \frac{1}{\sqrt{\left[1 - r^2\right]^2 + \left[2\zeta r\right]^2}} $$

We already know that \(\left[1 - r^2\right]^2 = (r^2 - 1)^2\), and \(r^2 - 1 > 1\). The term \(\left[2\zeta r\right]^2\) is always positive (or zero if \(\zeta=0\)).

Therefore, the entire denominator \(\sqrt{\left[1 - r^2\right]^2 + \left[2\zeta r\right]^2}\) will be greater than \(\sqrt{\left[1 - r^2\right]^2} = \left|1 - r^2\right| = r^2 - 1\).

Since the denominator is even larger than \(r^2 - 1\) (which is already greater than 1), the transmissibility \(TR = \frac{1}{\text{a value greater than } (r^2 - 1)}\) will be even smaller than in the undamped case. Consequently, the transmissibility will still be less than one.

Transmissibility and Vibration Isolation

When the frequency ratio \(\left(\frac{\omega}{\omega_n}\right)\) is greater than \(\sqrt{2}\), the vibrating system enters a region where it effectively isolates vibrations. This is a desirable operating range for vibration isolation systems because the transmitted force or motion is less than the input force or motion.

In summary, whether the system is damped or undamped, for values of the frequency ratio \(\left(\frac{\omega}{\omega_n}\right) > \sqrt{2}\), the transmissibility of the vibrating system will be less than one.