A damped system is modeled as illustrated in Figure 1.10....

A damped system is modeled as illustrated in Figure 1.10. The mass of the system is measured to be 5 kg and its spring constant is measured to be $5000 \mathrm{~N} / \mathrm{m}$. It is observed that during free vibration the amplitude decays to 0.25 of its initial value after five cycles, Calculate the viscous damping coefficient, $c$.

Key Concepts

Damping Ratio

The damping ratio is a dimensionless parameter that quantifies the relative magnitude of damping in a system compared to critical damping. It provides insight into whether the system is underdamped, critically damped, or overdamped, which in turn affects the decay rate of the oscillations.

Logarithmic Decrement

Logarithmic decrement is a measure of the rate at which oscillations decay in a damped system. It is defined as the natural logarithm of the ratio of consecutive amplitudes and is used to determine the damping ratio by relating the observed amplitude decay over a number of cycles to the dissipative properties of the system.

Natural Frequency

The natural frequency of an undamped system is the frequency at which the system would oscillate if there were no energy loss. It is determined by the mass and spring constant of the system, and serves as a reference point for assessing the effect of damping on the oscillatory behavior.

Viscous Damping

Viscous damping refers to a damping force that is directly proportional to the velocity of the oscillating system, characterized by the damping coefficient 'c'. This concept is critical in modeling and analyzing how resistive forces dissipate energy in dynamic mechanical systems.

Damped Harmonic Oscillator

A damped harmonic oscillator is a system where a mass subjected to a restoring force (typically provided by a spring) also experiences a resistive force (damping) that reduces the amplitude of oscillations over time. This concept forms the basis for analyzing systems that lose energy due to friction or other non-conservative forces.

Transcript

00:01 Okay, so we have a damp system that is modeled as illustrated in a figure 1 .10.

00:08 We have a mass of 5 kilograms and a spring constant k of 5 ,000 newtons per meter.

00:19 We have a free vibration amplitude decay to 0 .25 of its initial value after five cycles.

00:28 Five cycles.

00:31 We get 0 .25 of original or initial.

00:36 Calculate the viscous damping coefficient c, so that's what we want.

00:43 Okay, dokey.

00:46 Here is the science behind it.

00:49 For any two peak amplitudes of your peak or of that wave that are consecutive, aka one right after another, we can define e to the delta equals x0 over x1, which must equal.

01:24 X1 over x2, which must equal x2 over x3, which must equal x3 over x4, which must equal x4 over x5.

01:38 And in this situation, we have our initial, we have this guy, and we have that final, this guy.

01:46 So, we would say that e to the five deltas equals, see how everything cancels right here? see that? x0 over x5.

02:03 So let's go ahead and solve for delta.

02:09 To hear it of e, you're going to take the natural log of both sides, ln, ln.

02:14 So you get 5 times delta times ln of e equals the ln of x0 over x5.

02:23 But remember, what is this ratio? we decayed to 0 .25 of its original value.

02:29 So this is just ln of 4...

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