Damped motion and response time

A playground swing left alone slowly comes to rest, and a car that goes over a bump stops bouncing almost at once. Both are damped oscillations: resistive forces take energy out of the oscillation. How much damping there is decides whether the motion dies away slowly, settles quickly or creeps back, and a car’s suspension is designed to sit on the boundary between them.

What damping does

In a real oscillator, resistive forces such as air resistance, friction or the viscous drag of oil act against the motion. They do negative work on the oscillator, so its energy is gradually transferred to internal energy of the oscillator and its surroundings. This is damping. The amplitude decreases, because the energy of an oscillation is proportional to the square of its amplitude. The total energy is still conserved: it has moved to other stores, not been destroyed.

Light, critical and heavy damping

The graph shows the same oscillator, displaced by the same amount and released from rest, with three degrees of damping.

Damping after the same release from rest

Only the underdamped curve oscillates. Critical damping approaches equilibrium fastest among the non-oscillatory returns for the same mass, stiffness and initial conditions.

Scroll across the figure to read all labels.

Open full-size graph

What to noticeCalculated linear damping models, not measured traces. All start at x/A = 1 with zero velocity, the same mass and restoring stiffness, and no continuing drive. Time is scaled by the same undamped angular frequency ω₀. ζ labels relative damping; learners need to compare the curves, not memorise their solution formulas.

View figure data

Figure data: Damping after the same release from rest

Scaled time, ω₀t (unitless)	Under: ζ = 0.20	Critical: ζ = 1	Over: ζ = 2
0.00	1.000000	1.00000000	1.00000
0.05	0.998759	0.99879090	0.99883
0.10	0.995070	0.99532116	0.99561
0.15	0.988992	0.98981417	0.99072
0.20	0.980587	0.98247690	0.98447
0.25	0.969922	0.97350098	0.97712
0.30	0.957068	0.96306369	0.96889
0.35	0.942100	0.95132892	0.95996
0.40	0.925097	0.93844806	0.95047
0.45	0.906141	0.92456082	0.94055
0.50	0.885319	0.90979599	0.93029
0.55	0.862718	0.89427221	0.91979
0.60	0.838429	0.87809862	0.90911
0.65	0.812547	0.86137553	0.89830
0.70	0.785166	0.84419502	0.88742
0.75	0.756384	0.82664147	0.87650
0.80	0.726301	0.80879214	0.86558
0.85	0.695016	0.79071762	0.85467
0.90	0.662630	0.77248235	0.84381
0.95	0.629246	0.75414500	0.83300
1.00	0.594966	0.73575888	0.82226
1.05	0.559893	0.71737239	0.81161
1.10	0.524128	0.69902928	0.80105
1.15	0.487774	0.68076905	0.79059
1.20	0.450934	0.66262727	0.78024
1.25	0.413706	0.64463579	0.76999
1.30	0.376193	0.62682312	0.75986
1.35	0.338491	0.60921461	0.74984
1.40	0.300698	0.59183271	0.73994
1.45	0.262909	0.57469721	0.73016
1.50	0.225218	0.55782540	0.72049
1.55	0.187716	0.54123233	0.71095
1.60	0.150492	0.52493095	0.70153
1.65	0.113632	0.50893226	0.69222
1.70	0.077221	0.49324551	0.68304
1.75	0.041341	0.47787834	0.67397
1.80	0.006068	0.46283689	0.66502
1.85	-0.028521	0.44812592	0.65618
1.90	-0.062355	0.43374900	0.64746
1.95	-0.095365	0.41970851	0.63885
2.00	-0.127484	0.40600585	0.63036
2.05	-0.158653	0.39264146	0.62198
2.10	-0.188811	0.37961493	0.61371
2.15	-0.217906	0.36692510	0.60554
2.20	-0.245884	0.35457011	0.59749
2.25	-0.272701	0.34254748	0.58954
2.30	-0.298311	0.33085418	0.58170
2.35	-0.322676	0.31948669	0.57396
2.40	-0.345760	0.30844104	0.56632
2.45	-0.367532	0.29771287	0.55879
2.50	-0.387963	0.28729750	0.55135
2.55	-0.407029	0.27718991	0.54402
2.60	-0.424710	0.26738488	0.53678
2.65	-0.440991	0.25787693	0.52963
2.70	-0.455857	0.24866040	0.52259
2.75	-0.469301	0.23972948	0.51563
2.80	-0.481318	0.23107824	0.50877
2.85	-0.491905	0.22270064	0.50200
2.90	-0.501066	0.21459056	0.49532
2.95	-0.508805	0.20674184	0.48873
3.00	-0.515132	0.19914827	0.48222
3.05	-0.520059	0.19180364	0.47581
3.10	-0.523601	0.18470173	0.46948
3.15	-0.525778	0.17783633	0.46323
3.20	-0.526610	0.17120126	0.45706
3.25	-0.526122	0.16479038	0.45098
3.30	-0.524343	0.15859762	0.44498
3.35	-0.521301	0.15261694	0.43906
3.40	-0.517029	0.14684239	0.43321
3.45	-0.511562	0.14126808	0.42745
3.50	-0.504938	0.13588823	0.42176
3.55	-0.497195	0.13069711	0.41615
3.60	-0.488375	0.12568912	0.41061
3.65	-0.478521	0.12085875	0.40515
3.70	-0.467678	0.11620057	0.39975
3.75	-0.455893	0.11170929	0.39443
3.80	-0.443212	0.10737970	0.38918
3.85	-0.429686	0.10320672	0.38400
3.90	-0.415365	0.09918537	0.37889
3.95	-0.400299	0.09531077	0.37385
4.00	-0.384541	0.09157819	0.36888
4.05	-0.368143	0.08798299	0.36397
4.10	-0.351160	0.08452064	0.35912
4.15	-0.333643	0.08118674	0.35434
4.20	-0.315647	0.07797700	0.34963
4.25	-0.297227	0.07488723	0.34498
4.30	-0.278436	0.07191336	0.34039
4.35	-0.259328	0.06905145	0.33586
4.40	-0.239957	0.06629764	0.33139
4.45	-0.220375	0.06364819	0.32698
4.50	-0.200636	0.06109948	0.32262
4.55	-0.180791	0.05864798	0.31833
4.60	-0.160892	0.05629028	0.31409
4.65	-0.140988	0.05402305	0.30991
4.70	-0.121130	0.05184308	0.30579
4.75	-0.101366	0.04974725	0.30172
4.80	-0.081741	0.04773253	0.29771
4.85	-0.062304	0.04579601	0.29374
4.90	-0.043096	0.04393484	0.28983
4.95	-0.024163	0.04214628	0.28598
5.00	-0.005544	0.04042768	0.28217
5.05	0.012719	0.03877647	0.27842
5.10	0.030589	0.03719015	0.27471
5.15	0.048030	0.03566634	0.27105
5.20	0.065007	0.03420270	0.26745
5.25	0.081486	0.03279699	0.26389
5.30	0.097438	0.03144704	0.26038
5.35	0.112832	0.03015076	0.25691
5.40	0.127643	0.02890612	0.25349
5.45	0.141844	0.02771117	0.25012
5.50	0.155413	0.02656401	0.24679
5.55	0.168329	0.02546284	0.24351
5.60	0.180573	0.02440590	0.24027
5.65	0.192127	0.02339149	0.23707
5.70	0.202977	0.02241797	0.23391
5.75	0.213110	0.02148377	0.23080
5.80	0.222515	0.02058737	0.22773
5.85	0.231183	0.01972731	0.22470
5.90	0.239107	0.01890217	0.22171
5.95	0.246283	0.01811059	0.21876
6.00	0.252707	0.01735127	0.21585
6.05	0.258379	0.01662293	0.21297
6.10	0.263298	0.01592436	0.21014
6.15	0.267469	0.01525439	0.20734
6.20	0.270896	0.01461190	0.20458
6.25	0.273584	0.01399579	0.20186
6.30	0.275542	0.01340502	0.19917
6.35	0.276779	0.01283859	0.19652
6.40	0.277307	0.01229552	0.19391
6.45	0.277138	0.01177489	0.19133
6.50	0.276286	0.01127579	0.18878
6.55	0.274768	0.01079737	0.18627
6.60	0.272600	0.01033880	0.18379
6.65	0.269800	0.00989927	0.18134
6.70	0.266388	0.00947802	0.17893
6.75	0.262385	0.00907432	0.17655
6.80	0.257811	0.00868745	0.17420
6.85	0.252690	0.00831673	0.17188
6.90	0.247045	0.00796150	0.16959
6.95	0.240901	0.00762115	0.16734
7.00	0.234281	0.00729506	0.16511
7.05	0.227214	0.00698264	0.16291
7.10	0.219724	0.00668335	0.16075
7.15	0.211838	0.00639664	0.15861
7.20	0.203585	0.00612200	0.15650
7.25	0.194991	0.00585894	0.15441
7.30	0.186084	0.00560697	0.15236
7.35	0.176894	0.00536565	0.15033
7.40	0.167448	0.00513452	0.14833
7.45	0.157774	0.00491318	0.14636
7.50	0.147902	0.00470122	0.14441
7.55	0.137859	0.00449824	0.14249
7.60	0.127674	0.00430388	0.14059
7.65	0.117374	0.00411778	0.13872
7.70	0.106988	0.00393960	0.13687
7.75	0.096543	0.00376900	0.13505
7.80	0.086066	0.00360567	0.13325
7.85	0.075583	0.00344930	0.13148
7.90	0.065121	0.00329962	0.12973
7.95	0.054705	0.00315633	0.12800
8.00	0.044359	0.00301916	0.12630
8.05	0.034109	0.00288787	0.12462
8.10	0.023977	0.00276221	0.12296
8.15	0.013987	0.00264193	0.12133
8.20	0.004160	0.00252681	0.11971
8.25	-0.005483	0.00241664	0.11812
8.30	-0.014921	0.00231121	0.11655
8.35	-0.024136	0.00221031	0.11499
8.40	-0.033108	0.00211375	0.11346
8.45	-0.041821	0.00202136	0.11195
8.50	-0.050258	0.00193295	0.11046
8.55	-0.058403	0.00184836	0.10899
8.60	-0.066243	0.00176742	0.10754
8.65	-0.073763	0.00168997	0.10611
8.70	-0.080952	0.00161588	0.10470
8.75	-0.087798	0.00154500	0.10331
8.80	-0.094291	0.00147718	0.10193
8.85	-0.100423	0.00141231	0.10058
8.90	-0.106184	0.00135025	0.09924
8.95	-0.111569	0.00129088	0.09792
9.00	-0.116571	0.00123410	0.09661
9.05	-0.121185	0.00117978	0.09533
9.10	-0.125409	0.00112782	0.09406
9.15	-0.129238	0.00107813	0.09281
9.20	-0.132671	0.00103060	0.09157
9.25	-0.135709	0.00098514	0.09035
9.30	-0.138350	0.00094167	0.08915
9.35	-0.140597	0.00090009	0.08796
9.40	-0.142451	0.00086033	0.08679
9.45	-0.143916	0.00082231	0.08564
9.50	-0.144996	0.00078594	0.08450
9.55	-0.145695	0.00075117	0.08337
9.60	-0.146021	0.00071792	0.08227
9.65	-0.145978	0.00068613	0.08117
9.70	-0.145575	0.00065573	0.08009
9.75	-0.144820	0.00062667	0.07902
9.80	-0.143721	0.00059888	0.07797
9.85	-0.142289	0.00057231	0.07693
9.90	-0.140532	0.00054690	0.07591
9.95	-0.138463	0.00052262	0.07490
10.00	-0.136092	0.00049940	0.07390
10.05	-0.133431	0.00047720	0.07292
10.10	-0.130493	0.00045598	0.07195
10.15	-0.127290	0.00043570	0.07099
10.20	-0.123835	0.00041631	0.07005
10.25	-0.120143	0.00039777	0.06912
10.30	-0.116226	0.00038005	0.06820
10.35	-0.112099	0.00036312	0.06729
10.40	-0.107776	0.00034693	0.06639
10.45	-0.103273	0.00033146	0.06551
10.50	-0.098602	0.00031667	0.06464
10.55	-0.093781	0.00030253	0.06378
10.60	-0.088823	0.00028903	0.06293
10.65	-0.083743	0.00027611	0.06209
10.70	-0.078556	0.00026378	0.06126
10.75	-0.073278	0.00025198	0.06045
10.80	-0.067923	0.00024071	0.05964
10.85	-0.062506	0.00022994	0.05885
10.90	-0.057041	0.00021965	0.05807
10.95	-0.051544	0.00020982	0.05730
11.00	-0.046028	0.00020042	0.05653
11.05	-0.040507	0.00019144	0.05578
11.10	-0.034995	0.00018286	0.05504
11.15	-0.029506	0.00017466	0.05431
11.20	-0.024053	0.00016683	0.05358
11.25	-0.018648	0.00015934	0.05287
11.30	-0.013303	0.00015219	0.05217
11.35	-0.008032	0.00014535	0.05147
11.40	-0.002845	0.00013882	0.05079
11.45	0.002246	0.00013259	0.05011
11.50	0.007231	0.00012663	0.04944
11.55	0.012099	0.00012093	0.04879
11.60	0.016840	0.00011549	0.04814
11.65	0.021447	0.00011030	0.04750
11.70	0.025909	0.00010533	0.04686
11.75	0.030218	0.00010059	0.04624
11.80	0.034368	0.00009606	0.04563
11.85	0.038350	0.00009173	0.04502
11.90	0.042158	0.00008760	0.04442
11.95	0.045787	0.00008365	0.04383
12.00	0.049230	0.00007987	0.04324

Figure data: Damping after the same release from rest

Scaled time, ω₀t (unitless)	Under: ζ = 0.20	Critical: ζ = 1	Over: ζ = 2
0.00	1.000000	1.00000000	1.00000
0.05	0.998759	0.99879090	0.99883
0.10	0.995070	0.99532116	0.99561
0.15	0.988992	0.98981417	0.99072
0.20	0.980587	0.98247690	0.98447
0.25	0.969922	0.97350098	0.97712
0.30	0.957068	0.96306369	0.96889
0.35	0.942100	0.95132892	0.95996
0.40	0.925097	0.93844806	0.95047
0.45	0.906141	0.92456082	0.94055
0.50	0.885319	0.90979599	0.93029
0.55	0.862718	0.89427221	0.91979
0.60	0.838429	0.87809862	0.90911
0.65	0.812547	0.86137553	0.89830
0.70	0.785166	0.84419502	0.88742
0.75	0.756384	0.82664147	0.87650
0.80	0.726301	0.80879214	0.86558
0.85	0.695016	0.79071762	0.85467
0.90	0.662630	0.77248235	0.84381
0.95	0.629246	0.75414500	0.83300
1.00	0.594966	0.73575888	0.82226
1.05	0.559893	0.71737239	0.81161
1.10	0.524128	0.69902928	0.80105
1.15	0.487774	0.68076905	0.79059
1.20	0.450934	0.66262727	0.78024
1.25	0.413706	0.64463579	0.76999
1.30	0.376193	0.62682312	0.75986
1.35	0.338491	0.60921461	0.74984
1.40	0.300698	0.59183271	0.73994
1.45	0.262909	0.57469721	0.73016
1.50	0.225218	0.55782540	0.72049
1.55	0.187716	0.54123233	0.71095
1.60	0.150492	0.52493095	0.70153
1.65	0.113632	0.50893226	0.69222
1.70	0.077221	0.49324551	0.68304
1.75	0.041341	0.47787834	0.67397
1.80	0.006068	0.46283689	0.66502
1.85	-0.028521	0.44812592	0.65618
1.90	-0.062355	0.43374900	0.64746
1.95	-0.095365	0.41970851	0.63885
2.00	-0.127484	0.40600585	0.63036
2.05	-0.158653	0.39264146	0.62198
2.10	-0.188811	0.37961493	0.61371
2.15	-0.217906	0.36692510	0.60554
2.20	-0.245884	0.35457011	0.59749
2.25	-0.272701	0.34254748	0.58954
2.30	-0.298311	0.33085418	0.58170
2.35	-0.322676	0.31948669	0.57396
2.40	-0.345760	0.30844104	0.56632
2.45	-0.367532	0.29771287	0.55879
2.50	-0.387963	0.28729750	0.55135
2.55	-0.407029	0.27718991	0.54402
2.60	-0.424710	0.26738488	0.53678
2.65	-0.440991	0.25787693	0.52963
2.70	-0.455857	0.24866040	0.52259
2.75	-0.469301	0.23972948	0.51563
2.80	-0.481318	0.23107824	0.50877
2.85	-0.491905	0.22270064	0.50200
2.90	-0.501066	0.21459056	0.49532
2.95	-0.508805	0.20674184	0.48873
3.00	-0.515132	0.19914827	0.48222
3.05	-0.520059	0.19180364	0.47581
3.10	-0.523601	0.18470173	0.46948
3.15	-0.525778	0.17783633	0.46323
3.20	-0.526610	0.17120126	0.45706
3.25	-0.526122	0.16479038	0.45098
3.30	-0.524343	0.15859762	0.44498
3.35	-0.521301	0.15261694	0.43906
3.40	-0.517029	0.14684239	0.43321
3.45	-0.511562	0.14126808	0.42745
3.50	-0.504938	0.13588823	0.42176
3.55	-0.497195	0.13069711	0.41615
3.60	-0.488375	0.12568912	0.41061
3.65	-0.478521	0.12085875	0.40515
3.70	-0.467678	0.11620057	0.39975
3.75	-0.455893	0.11170929	0.39443
3.80	-0.443212	0.10737970	0.38918
3.85	-0.429686	0.10320672	0.38400
3.90	-0.415365	0.09918537	0.37889
3.95	-0.400299	0.09531077	0.37385
4.00	-0.384541	0.09157819	0.36888
4.05	-0.368143	0.08798299	0.36397
4.10	-0.351160	0.08452064	0.35912
4.15	-0.333643	0.08118674	0.35434
4.20	-0.315647	0.07797700	0.34963
4.25	-0.297227	0.07488723	0.34498
4.30	-0.278436	0.07191336	0.34039
4.35	-0.259328	0.06905145	0.33586
4.40	-0.239957	0.06629764	0.33139
4.45	-0.220375	0.06364819	0.32698
4.50	-0.200636	0.06109948	0.32262
4.55	-0.180791	0.05864798	0.31833
4.60	-0.160892	0.05629028	0.31409
4.65	-0.140988	0.05402305	0.30991
4.70	-0.121130	0.05184308	0.30579
4.75	-0.101366	0.04974725	0.30172
4.80	-0.081741	0.04773253	0.29771
4.85	-0.062304	0.04579601	0.29374
4.90	-0.043096	0.04393484	0.28983
4.95	-0.024163	0.04214628	0.28598
5.00	-0.005544	0.04042768	0.28217
5.05	0.012719	0.03877647	0.27842
5.10	0.030589	0.03719015	0.27471
5.15	0.048030	0.03566634	0.27105
5.20	0.065007	0.03420270	0.26745
5.25	0.081486	0.03279699	0.26389
5.30	0.097438	0.03144704	0.26038
5.35	0.112832	0.03015076	0.25691
5.40	0.127643	0.02890612	0.25349
5.45	0.141844	0.02771117	0.25012
5.50	0.155413	0.02656401	0.24679
5.55	0.168329	0.02546284	0.24351
5.60	0.180573	0.02440590	0.24027
5.65	0.192127	0.02339149	0.23707
5.70	0.202977	0.02241797	0.23391
5.75	0.213110	0.02148377	0.23080
5.80	0.222515	0.02058737	0.22773
5.85	0.231183	0.01972731	0.22470
5.90	0.239107	0.01890217	0.22171
5.95	0.246283	0.01811059	0.21876
6.00	0.252707	0.01735127	0.21585
6.05	0.258379	0.01662293	0.21297
6.10	0.263298	0.01592436	0.21014
6.15	0.267469	0.01525439	0.20734
6.20	0.270896	0.01461190	0.20458
6.25	0.273584	0.01399579	0.20186
6.30	0.275542	0.01340502	0.19917
6.35	0.276779	0.01283859	0.19652
6.40	0.277307	0.01229552	0.19391
6.45	0.277138	0.01177489	0.19133
6.50	0.276286	0.01127579	0.18878
6.55	0.274768	0.01079737	0.18627
6.60	0.272600	0.01033880	0.18379
6.65	0.269800	0.00989927	0.18134
6.70	0.266388	0.00947802	0.17893
6.75	0.262385	0.00907432	0.17655
6.80	0.257811	0.00868745	0.17420
6.85	0.252690	0.00831673	0.17188
6.90	0.247045	0.00796150	0.16959
6.95	0.240901	0.00762115	0.16734
7.00	0.234281	0.00729506	0.16511
7.05	0.227214	0.00698264	0.16291
7.10	0.219724	0.00668335	0.16075
7.15	0.211838	0.00639664	0.15861
7.20	0.203585	0.00612200	0.15650
7.25	0.194991	0.00585894	0.15441
7.30	0.186084	0.00560697	0.15236
7.35	0.176894	0.00536565	0.15033
7.40	0.167448	0.00513452	0.14833
7.45	0.157774	0.00491318	0.14636
7.50	0.147902	0.00470122	0.14441
7.55	0.137859	0.00449824	0.14249
7.60	0.127674	0.00430388	0.14059
7.65	0.117374	0.00411778	0.13872
7.70	0.106988	0.00393960	0.13687
7.75	0.096543	0.00376900	0.13505
7.80	0.086066	0.00360567	0.13325
7.85	0.075583	0.00344930	0.13148
7.90	0.065121	0.00329962	0.12973
7.95	0.054705	0.00315633	0.12800
8.00	0.044359	0.00301916	0.12630
8.05	0.034109	0.00288787	0.12462
8.10	0.023977	0.00276221	0.12296
8.15	0.013987	0.00264193	0.12133
8.20	0.004160	0.00252681	0.11971
8.25	-0.005483	0.00241664	0.11812
8.30	-0.014921	0.00231121	0.11655
8.35	-0.024136	0.00221031	0.11499
8.40	-0.033108	0.00211375	0.11346
8.45	-0.041821	0.00202136	0.11195
8.50	-0.050258	0.00193295	0.11046
8.55	-0.058403	0.00184836	0.10899
8.60	-0.066243	0.00176742	0.10754
8.65	-0.073763	0.00168997	0.10611
8.70	-0.080952	0.00161588	0.10470
8.75	-0.087798	0.00154500	0.10331
8.80	-0.094291	0.00147718	0.10193
8.85	-0.100423	0.00141231	0.10058
8.90	-0.106184	0.00135025	0.09924
8.95	-0.111569	0.00129088	0.09792
9.00	-0.116571	0.00123410	0.09661
9.05	-0.121185	0.00117978	0.09533
9.10	-0.125409	0.00112782	0.09406
9.15	-0.129238	0.00107813	0.09281
9.20	-0.132671	0.00103060	0.09157
9.25	-0.135709	0.00098514	0.09035
9.30	-0.138350	0.00094167	0.08915
9.35	-0.140597	0.00090009	0.08796
9.40	-0.142451	0.00086033	0.08679
9.45	-0.143916	0.00082231	0.08564
9.50	-0.144996	0.00078594	0.08450
9.55	-0.145695	0.00075117	0.08337
9.60	-0.146021	0.00071792	0.08227
9.65	-0.145978	0.00068613	0.08117
9.70	-0.145575	0.00065573	0.08009
9.75	-0.144820	0.00062667	0.07902
9.80	-0.143721	0.00059888	0.07797
9.85	-0.142289	0.00057231	0.07693
9.90	-0.140532	0.00054690	0.07591
9.95	-0.138463	0.00052262	0.07490
10.00	-0.136092	0.00049940	0.07390
10.05	-0.133431	0.00047720	0.07292
10.10	-0.130493	0.00045598	0.07195
10.15	-0.127290	0.00043570	0.07099
10.20	-0.123835	0.00041631	0.07005
10.25	-0.120143	0.00039777	0.06912
10.30	-0.116226	0.00038005	0.06820
10.35	-0.112099	0.00036312	0.06729
10.40	-0.107776	0.00034693	0.06639
10.45	-0.103273	0.00033146	0.06551
10.50	-0.098602	0.00031667	0.06464
10.55	-0.093781	0.00030253	0.06378
10.60	-0.088823	0.00028903	0.06293
10.65	-0.083743	0.00027611	0.06209
10.70	-0.078556	0.00026378	0.06126
10.75	-0.073278	0.00025198	0.06045
10.80	-0.067923	0.00024071	0.05964
10.85	-0.062506	0.00022994	0.05885
10.90	-0.057041	0.00021965	0.05807
10.95	-0.051544	0.00020982	0.05730
11.00	-0.046028	0.00020042	0.05653
11.05	-0.040507	0.00019144	0.05578
11.10	-0.034995	0.00018286	0.05504
11.15	-0.029506	0.00017466	0.05431
11.20	-0.024053	0.00016683	0.05358
11.25	-0.018648	0.00015934	0.05287
11.30	-0.013303	0.00015219	0.05217
11.35	-0.008032	0.00014535	0.05147
11.40	-0.002845	0.00013882	0.05079
11.45	0.002246	0.00013259	0.05011
11.50	0.007231	0.00012663	0.04944
11.55	0.012099	0.00012093	0.04879
11.60	0.016840	0.00011549	0.04814
11.65	0.021447	0.00011030	0.04750
11.70	0.025909	0.00010533	0.04686
11.75	0.030218	0.00010059	0.04624
11.80	0.034368	0.00009606	0.04563
11.85	0.038350	0.00009173	0.04502
11.90	0.042158	0.00008760	0.04442
11.95	0.045787	0.00008365	0.04383
12.00	0.049230	0.00007987	0.04324

• Light damping (underdamping): the body keeps oscillating, and the amplitude decreases gradually. The period stays almost the same.
• Critical damping: the body returns to equilibrium in the shortest possible time without oscillating, that is, without overshooting.
• Heavy damping (overdamping): the body does not oscillate, but it creeps back to equilibrium more slowly than with critical damping.

More damping is not always quicker. Beyond critical damping, extra damping slows the return.

Why critical damping matters

When a car goes over a bump, its springs start the body oscillating. Shock absorbers damp the oscillation. Close to critical damping, the car returns to its normal level quickly without bouncing up and down. Light damping would leave the car bouncing; heavy damping would make the suspension slow to recover, so the car would ride on compressed springs after a series of bumps. The pointers of analogue meters are damped close to critically for the same reason: they settle on the reading quickly without swinging past it.

Common mistakes

• Saying damping destroys energy; it transfers energy to internal energy.
• Thinking more damping always means a quicker return.
• Identifying critical damping only from the absence of oscillation.
• Comparing damping in two different systems released in different ways.