16361 Dynamics

Kinematics of a generalised rigid body moving in two dimesions

Case 1: System \( pxy \) is stationary in relation to system \( OXY \) and point \( p \) coincides with point \( O \)

$$ \bar{V}_A = \bar{V}_p + \bar{V}_{CIR} $$

$$ \dot{\bar{r}}_A = [ \dot{\bar{r}}_p + \bar{\omega} \times \bar{r} ] $$

$$ \bar{A}_A = \bar{A}_p + \bar{A}_{CIR} $$

$$ \ddot{\bar{r}}_A = [ \dot{\bar{r}}_p + \bar{\dot{\omega}} \times \bar{r} + \bar{\omega} \times (\bar{\omega} \times \bar{r}) ] $$

Case 2: System \( pxy \) is moving relative to system \( OXY \)

$$ \bar{V}_A = [\bar{V}_p + \bar{V}_{CIR}] + \bar{V}_{REL} $$

$$ \dot{\bar{r}}_A = [ \dot{\bar{r}}_p + \bar{\omega} \times \bar{r} ] + \dot{\bar{r}}_r $$

$$ \bar{A}_A = [\bar{A}_p + \bar{A}_{CIR}] + \bar{A}_{REL} + \bar{A}_{COR} $$

$$ \ddot{\bar{r}}_A = [ \dot{\bar{r}}_p + \bar{\dot{\omega}} \times \bar{r} + \bar{\omega} \times (\bar{\omega} \times \bar{r}) ] + \ddot{\bar{r}}_r + 2 \bar{\omega} \times \dot{\bar{r}}_r $$

\( \bar{V}_A \) and \( \bar{V}_p \); \( \bar{A}_A \) and \( \bar{A}_p \) are the absolute velocities and accelerations of points \( A \) and \( p \)

\( \omega \) is the angular velocity

\( r \) is the position vector

\( \bar{V}_{CIR} \) is the circular component as defined by \( \bar{V}_{CIR} = \bar{\omega} \times \bar{r} \)

\( \bar{A}_{CIR} \) is the circular component of acceleration as defined by \( \bar{A}_{CIR} = \bar{A}_{tangental} + \bar{A}_{normal} = \dot{\bar{\omega}} \times \bar{r} + \bar{\omega} \times ( \bar{\omega} \times \bar{r} ) \)

\( \bar{V}_{REL} \) and \( \bar{A}_{REL} \) are the relative velocity and acceleration of \( A \) across the body

\( \bar{A}_{COR} \) is the Coriolis component of acceleration, defined by \( \bar{A}_{COR} = 2 \bar{\omega} \times \dot{\bar{r}}_r \)

Free Vibration

System can be described by a generalised equation

$$ m \ddot{x} + c \dot{x} + kx = 0 $$

The solution will be in the form

$$ x = e^{- \zeta \omega_n t} ( A \sin \omega_d t + B \cos \omega_d t ) $$

Where \( \zeta \) is the damping ratio, defined by \( \zeta = \frac{c}{c_c} = \frac{actual damping}{critical damping} \)

\( \omega_n \) and \( \omega_d \) are the natural and dampened angular frequencies in rad/s respectively

The critical damping coefficient can be found by

$$ c_c = 2 \sqrt{km} $$

or

$$ c_c = 2 m \omega _n $$

Natural Frequency

The natural frequency can be found with the mass of the system and the spring constant, \( k \)

$$ \omega _n = \sqrt{ \frac{k}{m} } $$

This can be converted to Hertz via

$$ f = \frac{\omega _n}{2 \pi} $$

The dampened frequency can be found via the damping ratio

$$ \omega_d = \omega_n \sqrt{1 - \zeta^2} $$

The logarithmic decrements of the amplitude due to damping, \( \delta\) can be found by

$$ \frac{\hat{x}_p}{\hat{x}_{p+n}} = e^{n \delta} $$

Where \( \hat{x}_p \) is the amplitude of a peak and \( \hat{x}_{p+n} \) is the amplitude of the peak succeeding it.

Alternatively, the logarithmic decrement can be found by

$$ \delta= \frac{2 \pi \zeta}{\sqrt{1- \zeta^2}} $$

For underdamped systems only, the logarithmic decrement is also related to the damping ratio

$$ \zeta = \frac{\delta}{2 \pi} $$

Forced Vibration

A system excited by a harmonic external force can be described by

$$ m \ddot{x} + c \dot{x} + kx = F_0 \cos \omega t $$

For which, the steady state solution is

$$ x = X \cos ( \omega t - \Phi ) $$

Where

$$ X = \frac{\frac{F_0}{k}}{\sqrt{(1- \beta^2)^2+(2 \zeta \beta)^2}} $$

The phase difference, \( \Phi \) can be determined by

$$ \tan \Phi = \frac{2 \zeta \beta}{1 - \beta ^2} $$

or

$$ \beta = \frac{\omega}{\omega _n} $$

Where \( \omega \) is the forcing angular frequency

\( \beta \) is the frequency ratio

Other Spring Systems

For springs in series, the equivalent stiffness obeys the inverse sums rule

$$ \frac{1}{k_{total}} = \frac{1}{k_1} + \frac{1}{k_2} $$

For springs in parallel, their total stiffness is simply additive

$$ k_{total} = k_1 + k_2 $$

Transmitted Vibration

The transmissibility of a system is a ratio of the output to input force or displacement

$$ TransmittedForce = c \dot{x} + kx $$

$$ TR = \frac{F_t}{F_o} = \frac{\sqrt{1+ (2 \zeta \beta)^2}}{\sqrt{(1 - \beta^2)^2+(2 \zeta \beta)^2}} $$

Vibration Measurement


Where x is the displacement to be measured, y is the displacement of the seismic mass, z is the displacement actually measured.

$$ \left|\frac{z}{x} \right| = \frac{\beta^2}{\sqrt{(1 - \beta^2)^2+(2 \zeta \beta)^2}} $$