# 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.

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