Logarithmic decrement

Key takeaways

Logarithmic decrement belongs to science; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Logarithmic decrement to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Logarithmic decrement from memory before moving on to harder problems.

Reference excerpt

Logarithmic decrement, δ {\displaystyle \delta } , is used to find the damping ratio of an underdamped system in the time domain. The method of logarithmic decrement becomes less and less precise as the damping ratio increases past about 0.5; it does not apply at all for a damping ratio greater than 1.0 because the system is overdamped.

Method The logarithmic decrement is defined as the natural log of the ratio of the amplitudes of any two successive peaks:

δ = 1 n ln ⁡ x ( t ) x ( t + n T ) {\displaystyle \delta ={\frac {1}{n}}\ln {\frac {x(t)}{x(t+nT)}}}

where x(t) is the overshoot (amplitude - final value) at time t and x(t + nT) is the overshoot of the peak n periods away, where n is any integer number of successive, positive peaks. The damping ratio is then found from the logarithmic decrement by:

ζ = δ 4 π 2 + δ 2 {\displaystyle \zeta ={\frac {\delta }{\sqrt {4\pi ^{2}+\delta ^{2}}}}}

Thus logarithmic decrement also permits evaluation of the Q factor of the system:

Q = 1 2 ζ {\displaystyle Q={\frac {1}{2\zeta }}}

Q = 1 2 1 + ( n 2 π ln ⁡ x ( t ) x ( t + n T ) ) 2 {\displaystyle Q={\frac {1}{2}}{\sqrt {1+\left({\frac {n2\pi }{\ln {\frac {x(t)}{x(t+nT)}}}}\right)^{2}}}}

The damping ratio can then be used to find the natural frequency ωn of vibration of the system from the damped natural frequency ωd:

ω d = 2 π T {\displaystyle \omega _{d}={\frac {2\pi }{T}}}

ω n = ω d 1 − ζ 2 {\displaystyle \omega _{n}={\frac {\omega _{d}}{\sqrt {1-\zeta ^{2}}}}}

where T, the period of the waveform, is the time between two successive amplitude peaks of the underdamped system.

Simplified variation The damping ratio can be found for any two adjacent peaks. This method is used when n = 1 and is derived from the general method above:

ζ = 1 1 + ( 2 π ln ⁡ ( x 0 x 1 ) ) 2 {\displaystyle \zeta ={\frac {1}{\sqrt {1+\left({\frac {2\pi }{\ln \left({\frac {x_{0}}{x_{1}}}\right)}}\right)^{2}}}}}

where x0 and x1 are amplitudes of any two successive peaks. For system where ζ ≪ 1 {\displaystyle \zeta \ll 1} (not too close to the critically damped regime, where ζ ≈ 1 {\displaystyle \zeta \approx 1} ).

… excerpt ends here. Continue reading the full article.

Illustrations

Worked examples

Example 1 — a first encounter with Logarithmic decrement

Start with the simplest possible case. Write down what Logarithmic decrement claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In science, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Logarithmic decrement before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Logarithmic decrement ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Logarithmic decrement

In research: Logarithmic decrement appears in science research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Logarithmic decrement in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Logarithmic decrement is common in secondary-school and first-year university syllabi. It links to neighbouring topics Kinematic properties, Logarithms, so understanding it makes those chapters shorter.
In everyday life: Look for Logarithmic decrement outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

How to study Logarithmic decrement in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Logarithmic decrement means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Logarithmic decrement out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Logarithmic decrement in simple terms?

Logarithmic decrement, δ {\displaystyle \delta } , is used to find the damping ratio of an underdamped system in the time domain. The method of logarithmic decrement becomes less and less precise as the damping ratio increases past about 0.5; it does not apply at all for a damping ratio greater tha…

Why does Logarithmic decrement matter?

Because it connects several science ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Logarithmic decrement?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Logarithmic decrement.