5.4.5.1. A simple system with 1 DOF

Figure 5.20 Example model

• DOF is 1 and EOM is \(m\ddot{x}+c\dot{x}+kx=0\).

• Sphere, translational joint and spring force

• No gravity ( or Simulates both static analysis and Eigenvalue analysis)

Under damped case

• \(m=4.084594(kg),c=1(N\sec /mm),k=100(N/mm)\)

Table 5.2 Result

Expression	Result
Undamped natural frequency (Hz)	2.490266E+01
Real of damped Eigenvalue (/2PI)	-1.948235E+01
Imaginary of damped Eigenvalue(Hz)(Damped frequency)	+/- 1.551066E+01
Damping ratio	7.823400E-01

Critical damped case

• Can compute the critical damping.

• \(m=4.084594(kg),c=1.2782(N\sec /mm),k=100(N/mm)\)

  \(\zeta =\frac{C}{{{C}_{C}}}=0.78234=\frac{1}{Cc}\)

  \({{C}_{C}}\approx 1.2782`\)

Table 5.3 Result

Expression	Result
Undamped natural frequency (Hz)	2.490266E+01
Real of damped Eigenvalue (/2PI)	-2.490233E+01
Imaginary of damped Eigenvalue(Hz)(Damped frequency)	+/- 1.268287E-01
Damping ratio	9.999870E-01

• Damping ratio is approximated 1.0.

Over damped case

• In over damped case damping coefficient is greater than critical damping coefficient.

• \(m=4.084594(kg),c=2(N\sec /mm),k=100(N/mm)\)

Table 5.4 Result

Mode	Expression	Result
1	Undamped natural frequency (Hz)	0.000000E+00
	Real of damped Eigenvalue (/2PI)	-6.893309E+01
	Imaginary of damped Eigenvalue (Hz) (Damped frequency)	0.000000E+00
	Damping ratio	1.000000E+00
2	Undamped natural frequency (Hz)	0.000000E+00
	Real of damped Eigenvalue (/2PI)	-8.996294E+00
	Imaginary of damped Eigenvalue (Hz) (Damped frequency)	0.000000E+00
	Damping ratio	1.000000E+00

• In over damped case, damped frequency is all zero (0.0 Hz). Because, damped Eigenvalue has only two real values in the same mode shape.

• And reported undamped natural frequency and damping ratio have no meaning. Because, undamped natural frequency is analytically computed using critical and under damped case definition. So, when number of reported modes are greater than DOF of system, check the over damped system exists using under damped type.