This was all necessary to investigate as sometimes objects/structures such as cars will experience motions which bring them close to their natural frequency of vibration, be It sinusoidal or otherwise which causes the amplitude experienced by the object to reach Its maximum which can have significant effects on that object and the factors affecting it during oscillation. Theory There are numerous equations involved in this experiment all of which are important In different ways and are listed below: Equation 1 The first equation of motion for the carnage Is: mix.. Xx. Xx= moonshines Where: m – Mass of carriage c – Damping comment k – Spring stiffness – Frequency of input vibrations (Roads/s) MO – Mass of the out-of-balance rotating masses e – Eccentricity of these masses This is the standard equation of motion for single degree of freedom systems that are generally in the setup of the following diagram: However, this specific experiment is with respect to rotating out-of-balance excitation in the system.
This means that the right hand side of the equals in the equation is what the equation of motion of this single degree of freedom system equates to with aspect the initial conditions/conditions placed in this experiment, e. G. The mass equates to MO therefore mix.. On the left hand side of the equation turns into the MO as the rest has equated into. Due to this single degree of freedom systems setup, oscillations that affect the system register on this equipment as it is so sensitive, for example in car suspensions when they absorb shocks.
Furthermore, this equation has been equated to specific parameters such as the coefficient of the sign being Moe which is the amplitude of the sin wave- This amplitude being affected by the ass placed in the system (MO), the eccentricities of the masses (e) and the Truculence AT Input violations AT ten system (w), wanly all erect ten residual equator. Besides this, the external forces are coming from this spring (mix.. ) which means that we have rearranged a prior equation (mix.. + Xx. + Xx= moonshines) to find equation 1 as the mix.. Was the subject of the equation until it was altered. Equation 2 x.. Gown x. + Wynn = [Moe/m]sinew writ = (k/m)1/2 – Untapped natural frequency q – Damping ratio This equation has been found by rewriting Equation 1 and has some of the armaments changed, most noticeably involving the introduction of the damping ratio. The damping ratio can be described as a dimensionless measurement that explains how oscillations in a system decay after a disturbance has occurred. [l] This shows that the damping ratio in the equation is multiplied by the natural frequency of the system and the damping constant as these are all factors that involve oscillations and some form of damping.
This involves on = (k/m)1/2 which explains that the natural frequency is dictated by the root of the spring constant over the mass in the system s this is the cause for oscillations in the system when a force is applied causing the spring to move up and down and hence oscillate. The purpose of rearranging this equation is to find the wave of oscillation of the system which is [Moe/m]sinew, hence the left side of this equation shows that the wave construction of the system depends on the mass in the system and also the damping factors like the damping ratio etc.
Equation 3 x = ox stun(wet-9) I Nils equator Is In terms AT ten Minimal contraltos as It Involves xx wanly according to the equation below where ox is the subject of the function which involves the initial ass (MO) of the system, the frequency of vibrations of the system, the actual mass during oscillation in the system (m) the damping ratio (q), and the eccentricity of these masses. This equation is used as a whole to determine the amplitude and nature of the sin wave.
This is involving the same factors as equation 2, but is Just in general terms of x (Displacement), whereas equation 2 is in terms of the acceleration (x.. ) and velocity (x. ) in the system according to the mass and damper in the system respectively. The bracket after sin involving wet and cap is the angle of the sign wave ND determines how the wave functions but is determined by a combination of these two factors from the system.
This factor involving cap can be found using the following equation: Where we need to rearrange this equation by taking tan-I of the right hand side of the equation to make cap the subject and place it into the ox equation so that we have all the necessary information to find our equation 3 answer, but only after we have found our damping ratio through the equation: By dividing both sides of this equation by 2, we can find the damping ratio and finally omelet equation 3 to find the necessary answer we need.
Loquacious theory We have the equation: X = ox sin(wax + Ox) Y = you sin(wet + Ay) winner u Is ten Key Doctor name Tort ten Alehouses Deluge as tens Is ten apneas delay of the system response and is an objective that the Loquacious figure aims to find.
By introducing signals from the engine and by factoring in signals from the system, these equations can changed into the following necessary equations for use in the Loquacious figure to find C]: (When sin 0 = c YMCA = you = d Therefore we can find the key equation: TN o = cold And we can finally take the inverse of sign to the right hand side of the equation to give us the full equation we need for use in the Loquacious figure: = sin-I(c/d) Where 0 is the phase delay of the system response.
Description AT system The system consisted of a fairly complex setup which involved the following: – Moving belts and wheels causing the system to oscillate – A motor that moved the belts and wheels that oscillate the system – A laser that projects the oscillation – A white wall that the laser that is projecting the oscillation shines on A mirror that reflects the laser in such a way that it causes the oscillations to change behavior by going from oscillating in the up and down direction, to reflecting in a circular manner that creates the Loquacious circle, BUT only when it is flipped into the right position – A machine that controls the frequency of oscillations that the motor/system outputs – Gauges that tell you how fast the motor is moving in revolutions per minute – Dampers, springs, and out of balance masses that are the core components of the system This is displayed in the following setup diagram: