Springs

Simple hooke’s law for linear springs:

\[F = k\delta\]

Where

• \(F\) is force
• \(k\) is stiffness of spring
• \(\delta\) is deformation of spring

Springs in parallel:

\[k_{eq} = \sum k_i\]

Springs in series:

\[\frac{1}{k_{eq}} = \sum \frac{1}{k_i}\]

aka same as capacitors.

Resonance frequency

q	Free	Forced (cantilever)	Torsion
Resonance frequency \(\omega\)	\(\sqrt{\frac{K}{I}} = \sqrt{\frac{g}{l}}\)	\(\sqrt{\frac{K}{m}}\)	\(\sqrt{\frac{K}{I}}\)
Stiffness \(K\)	\(mgl\)	\(\frac{3 E I}{L^3}\)	\(\frac{GJ}{L}\)

Free undamped system

Frequency:

\[\omega = \sqrt{\frac{K}{m}}\]

Where

• \(\omega\) is natural resonance frequency
• \(K\) is stiffness of object
• \(m\) is mass of object

\[K = mgl\]

Where

• \(m\) is mass of object
• \(g\) is gravity
• \(l\) is length of spring

Response:

\[x(t) = A_o \sin{\omega t + \phi_o}\]

Where

• \(A_o\) is amplitude of oscillation
• \(\omega\) is natural resonance frequency
• \(t\) is time
• \(\phi_o\) is phase (initial position, for \(t = 0\))

Forced undamped system, 1 degree of freedom, cantilever

Frequency:

\[\omega = \sqrt{\frac{K}{M}}\]

Where

• \(\omega\) is natural resonance frequency
• \(K\) is stiffness of object
• \(M\) is mass of object

\[K = \frac{3 E I}{L^3}\]

Where

• \(E\) is Young’s Modulus
• \(I\) is area moment of inertia
• \(L\) is length of beam/spring

Frequency for damped systems

\[f_d = f_n \sqrt{1 - \zeta^2}\]

Where

• \(f_d\) is damped natural frequency
• \(f_n\) is undamped natural frequency
• \(\zeta\) is damping ratio

Torsion

\[\omega = \sqrt{\frac{K_t}{I}}\]

Where

• \(\omega\) is natural resonance frequency
• \(K_t\) is torsional stiffness of object
• \(I\) is mass moment of inertia

\[K_t = \frac{GJ}{L}\]

Where

• \(G\) is shear modulus
• \(J\) is polar moment of inertia
• \(L\) is length of spring/shaft