# 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.

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