KIN 840
2006-1
Dynamics pt. 2: Equations of
motion and system
identification
© 2006 by Stephen Robinovitch, All Rights Reserved
Outline
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Lagrange's equation
Kinetic energy
Potential energy
Inverted pendulum model
Mass-spring model
Characteristic equations
Free vibration response
Damping
Equations of motion (2D):
Inverse dynamics models:
estimate joint torques from
applied forces and segment
motions
Forward dynamics models:
estimate segment motions
from joint torques and
applied forces
Linear and nonlinear differential equations
• Linear equations have constant
coefficients, or coefficients that
involve the independent variable
(e.g., time). There are wellestablished linear analysis
techniques.
• Nonlinear equations have
powers or other functions or
products of the dependant
variable and its derivatives.
Analysis often involves
linearization, or solution via
numerical integration.
Deriving equations of motion with Lagrange’s
Equation
Lagrange’s Equation (cont)
Kinetic energy of a rigid body (general)
Example. Rotation about a fixed point
Example. Kinetic energy of a pendulum
Gravitational potential energy
Elastic potential energy
Example. Inverted pendulum with distributed
mass
Example. Hanging pendulum
• Use both the momentum
technique and Lagrange’s
equation to determine the
equation of motion
(EOM) for the pendulum
at right.
• Compare your EOM to
that for the inverted
pendulum. What indicates
the difference in system
stability?
Example. Mass-spring vibratory system
EOM for the mass-spring system
Characteristic equation: undamped system
Solution to the EOM for the mass-spring
system
Example. Determining leg stiffness from
measured natural frequency of vibration
Cavagna (1976) used a massspring representation of the body
to estimate the stiffness of the leg
on landing from a jump
The measured natural frequency
was 2.86 Hz (or 17.9 rad/s) for an
individual of 76 kg body mass
Neglecting damping, what is the
estimated leg stiffness in N/m for
this individual?
Effect of damping on system response
Characteristic equation: damped system
Underdamped, critically damped, and
overdamped systems
Damped free vibration
Logarithmic decrement
Example. System identification in damped
free vibration.
(a) For the unforced, damped vibratory system shown below, define the
following in terms of the parameters shown (and their time derivatives):
(i) equation of motion, (ii) undamped natural frequency ωn, (iii)
damping ratio ζ, (iv) damped natural frequency ωd.
(b) Determine the stiffness k (in N/m) and damping b (in Nm/s), if the time
between successive peaks in x is 1 sec, x0/ x1 = 2, and m = 70 kg.
m
k
x
b