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Logarithmic decrement measures amplitude loss between successive same-sign peaks of an underdamped oscillation.
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Formula
δ dimensionless; x in any consistent length unit
| Symbol | Meaning | Unit |
|---|---|---|
| Logarithmic decrement per period | dimensionless | |
| Magnitudes of successive same-sign oscillation peaks | m |
\delta=\ln\left(\frac{x_k}{x_{k+1}}\right)For linear viscous underdamping the envelope decays exponentially, making successive peak ratios ideally constant. Use magnitudes of same-sign peaks, not adjacent positive and negative extrema. Nonlinear or Coulomb damping can make decrement amplitude-dependent.
For linear viscous underdamping the envelope decays exponentially, making successive peak ratios ideally constant. Use magnitudes of same-sign peaks, not adjacent positive and negative extrema. Nonlinear or Coulomb damping can make decrement amplitude-dependent.
Logarithmic decrement measures amplitude loss between successive same-sign peaks of an underdamped oscillation.
Base equation for the target quantity
Preserve signs, boundary conditions and units when rearranging.
Work through the numerical example for “Logarithmic decrement of a damped oscillation” and check its unit and magnitude.
If peak amplitude falls from 10 mm to 8 mm, δ=ln(10/8)≈0.223 per period.
Substituting a related quantity that shares the same symbol.
Use variable definitions and check: δ dimensionless; x in any consistent length unit.
Formula cluster
Logarithmic decrement measures amplitude loss between successive same-sign peaks of an underdamped oscillation.
If peak amplitude falls from 10 mm to 8 mm, δ=ln(10/8)≈0.223 per period.
Estimate damping from a free-decay experiment.
Here is how to work through a typical Logarithmic decrement of a damped oscillation (δ = ln(xk/xk+1)) task step by step:
Task
Work through the numerical example for “Logarithmic decrement of a damped oscillation” and check its unit and magnitude.
Solution path
If peak amplitude falls from 10 mm to 8 mm, δ=ln(10/8)≈0.223 per period.
Two positive peaks are one period apart; an adjacent negative peak is only half a period away. For decrement per period, compare same-sign peaks. Use consecutive same-sign peaks from a linear underdamped free response. Application: Estimate damping from a free-decay experiment. The numerical example If peak amplitude falls from 10 mm to 8 mm, δ=ln(10/8)≈0.223 per period anchors the idea in a concrete measurement case. The corresponding units are δ dimensionless; x in any consistent length unit. If peak amplitude falls from 10 mm to 8 mm, δ=ln(10/8)≈0.223 per period. This checks that the reference quantities are compatible.
With an exponential envelope, amplitude ratio per period is constant. The logarithm turns that ratio into an additive damping measure. Larger δ means more decay per cycle, and the dimensionless value is obtained directly from measured peaks. Use consecutive same-sign peaks from a linear underdamped free response. The corresponding units are δ dimensionless; x in any consistent length unit. If peak amplitude falls from 10 mm to 8 mm, δ=ln(10/8)≈0.223 per period. This checks that the reference quantities are compatible.
Compare peaks separated by several cycles and divide by their count: δ=(1/N)ln(xk/xk+N). This averages measurement noise if the decay model fits. Account for offsets, outliers and overlapping modes. Use consecutive same-sign peaks from a linear underdamped free response. Scope: For linear viscous underdamping the envelope decays exponentially, making successive peak ratios ideally constant. Use magnitudes of same-sign peaks, not adjacent positive and negative extrema. Nonlinear or Coulomb damping can make decrement amplitude-dependent. A practical application is Estimate damping from a free-decay experiment.
Write the quantity and unit from the variable list next to every value: δ dimensionless; x in any consistent length unit. Substitute only after converting to a consistent unit system, and state sign or reference direction explicitly. The example If peak amplitude falls from 10 mm to 8 mm, δ=ln(10/8)≈0.223 per period. is a calculation check: result, unit and order of magnitude must fit the described situation. Round after intermediate calculations so small contributions are not lost prematurely. Use consecutive same-sign peaks from a linear underdamped free response.
The background states this starting condition: For linear viscous underdamping the envelope decays exponentially, making successive peak ratios ideally constant. Use magnitudes of same-sign. That limits when the result is a valid model prediction. Compare the problem’s boundary conditions with this assumption before substituting values; real components, material properties or loads can introduce additional effects. The cited reference supports the relation used here but does not remove the need to check whether the specific case fits the model. Use consecutive same-sign peaks from a linear underdamped free response.
Quanta exam set
Curated exam set for "Logarithmic decrement of a damped oscillation":
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Which formula describes Logarithmic decrement of a damped oscillation?
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How do you rearrange Logarithmic decrement of a damped oscillation for Base equation for the target quantity?
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Which common mistake happens with Logarithmic decrement of a damped oscillation?
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