View Full Version : Spring & Damping Constants

Dr M. C. Siff
07-11-1997, 01:15 AM
Rosemary Reid wrote:

> In a recent reply aboutimpact spring parameters there was a reply saying
> that the spring constant k varied over time. Does anybody have a reference
> for both the spring constant k and damping constant c for the lower mainly
> but also upper limbs as it changes over time during locomotion?

In my PhD (Ballistic Analysis of Human Knee Stability, 1986,
University of the Witwatersrand)), I studied these factors for the knee joint
and found:

MALES (93 young adult subjects):

Mean stiffness k = 6750 N/m (SD = 1763)
Mean damping ratio v = 0.117 (SD = 0.031)

The stiffness and damping ratio both increased with bodymass.

FEMALES (18 young adult):

Mean stiffness k = 5346 N/m (SD = 835)
Mean damping ratio v = 0.136 (SD = 0.037)


After free standing squats without weights, the mean damping ratio
after exerciseincreased by 14%, while the stiffness decreased by 6%. After
squatting with a load of about 60% of bodymass, these values changed
by +17% and - 9%, respectively.

The damping ratio increased even more markedly after impulsive
'plyometric' style jumps (23%). in a group of female gymnasts that I

If the mechanical characteristics of the subjects were studied during
loaded squat bounces the damping ratio and the siffness both

These results concurred with similar work done by:

Greene & Mc Mahon T (1979) Reflex stiffness of man's anti-gravity
muscles during knee-bends while carrying extra weights J Biomechs
12: 881-891.

It is also well documented that increase in muscle activity and axial
loading on a joint increases its stiffness:

Markolf K et al (1976) Stiffness and laxity of the knee J Bone &
Joint Surg 58A: 583-593

Crowninshield R et al (1976) An analytical model of the knee J
Biomechs 9: 397-405

Hsieh & Walker (1976) Stabilising mechanisms of the loaded and
unloaded knee joint J Bone Jt Surg 58A: 87-93


My thesis also briefly examined the ankle joint, which exhibited the
following characteristics of young males:

Stiffness k = 9515 N/m
Damping ratio v = 0.148

These values are similar to those obtained from in vitro methods
(0.20) and from theoretical models by other researchers (Cavagna
(1970); Elastic bounce of the body J Appl Physiol 29: 279-282; Bach
et al (1983) Mechanical resonance of the human body during voluntary
oscillations about the ankle joint J Biomechs 16, 1: 85-90).


Relaxed forced oscillation methods by Crowninshield et al (1975) The
impedance of the human knee J Biomech 9: 529-535, and Doriu L & Hull
M (1984) Dynamic simulation of the leg in torsion J Biomech 17, 1:
1-9 produced the following results:

Varus-valgus stiffness at 0 degrees of flexion: 9800 - 13800 N/m
Varus-valgus stiffness at 30 degs of flexion: 6800 - 8800 N/m
Varus-vagus damping ratio : 0.12 - 0.25
Rotational stiffness at 15 degs of flexion: 6000 - 11 000 N/m

A large decrease in stiffness for both modes of oscillation after
reconstructive surgery of ligaments or joint capsule.

Greene & Mc Mahon (cited earlier) had subjects execute 20 small
oscillations while in an isometric holding position on a spruce wood
springboard, and found that stiffness is a function of knee angle and
that damping ration was 0.34 for the range of knee angles studied.
They also found that the damping ratio for running is 0.55,
considerably higher than the value obtained for the relaxed
oscillatory state analysed by myself and others. It is evident that
the mechanical characteristics of the joints alters with degree of
muscle tension, joint angle, levels of fatigue and so forth.

No doubt there are more recent studies, but I trust that the above
information will prove useful.


Dr Mel C Siff
School of Mechanical Engineering
University of the Witwatersrand
WITS 2050 South Africa