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EEEN30150: Modelling and Simulation - Dynamic Equations - Engineering Assignment Help
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Subject Code: EEEN30150
The input voltage signal is a single phase, 230 V (rms), 50 Hz sinusoid denoted “Mains Voltage” in Figure 2.1. Assume the four signal diodes in the bridge rectifier to be identical. Assume that they can be reasonably approximated by an ideal diode in series with a small bulk resistance, i.e. for a (forward bias) voltage v across one of these diodes the resulting (forward) current i flowing through 
. Is is the reverse saturation current, n is the ideality factor, V T is the thermal voltage (= kT/q where k is Planck’s constant, q is the magnitude of the electron charge and T is the temperature in K, i.e. in degrees Kelvin) and R Bulk is the bulk resistance. Appropriate parameter values for this model of the signal diode ar 
Assume a temperature of 300 K. By any means you choose determine a piecewise linear approximation to the DC characteristic of the signal diodes and employ this approximation in your simulation. In order to do this well you will need to make some credible estimate of the maximum current which could be flowing through a signal diode in forward bias. Your piecewise linear approximation need only be and should only be valid over a current range up to perhaps twice this value.
To keep things somewhat simple assume an ideal transformer with a turns ratio of 20:1. Again to keep things slightly simple model the Zener diode in the reverse bias region as an ideal Zener diode V Lwith a Zener breakdown voltage of 6.5 V in series with a very low resistance in the breakdown region of just 5.3 Ohm.
The electrolytic smoothing capacitor is rather large, C ? 2 , 200 ? F being the nominal capacitance. The tolerance of this capacitor is 20% The resistor R s is a 10% tolerance, 270 ? nominal, 1?2 W carbon film resistor. The load being supplied by the power supply is modelled by its Thévenin equivalent circuit. As the load generates no power this equivalent circuit consists of an input impedance but no voltage source. The load has several modes, depending on which devices are on or off and accordingly the impedance varies. There are two devices attached. One offers an inductive load and one a purely resistive load. Accordingly the load can be modelled as indicated in Figure
Derive equations describing the performance of the circuit for all four modes of the load (namely both devices off, both devices on and one device on. You may assume that the power supply is switched on at time t = 0, but that it has been off for a reasonable amount of time prior to this.
Accordingly, the voltage across the smoothing capacitor and the current through the inductor in the load have both had the time to decay to zero so that they may both be assumed to equal zero at time t = 0. For each of the four modes determine the resulting load voltage (denoted V L in Figure 2.1) for several cycles of the input voltage (sufficiently many that transients have passed). Given the precision of the smoothing capacitor and the resistance R S what range of steady-state load voltages do you expect to see? What is the maximum power dissipated in resistor R S ? Do you think that a 1?2
W power rating for this resistor is high enough?
For top credit (i.e. A or A+ grade) determine the load voltage for the following scenario: the power supply is switched on at time t = 0 at which time the smoothing capacitor voltage and inductor current are zero, neither attached loading device is active at this time; after 2 sec the resistive load is switched on; after a further 2.2 sec the inductive load is switched on and it remains active for the next 1.263 sec; subsequently the inductive device is switched off and remains off.
Problem 2: Vehicle Suspension
A vehicle is assumed to be completely symmetric about its central axis, i.e. the left hand side is the exact mirror image of the right hand side. A model for the suspension system of half of the vehicle is shown in Figure 3. The following parameter values have been deemed appropriate:
The wheelbase a+b and weight are approximately the values for a sixth generation BMW 5-series. Note that in the model half of the sprung mass, i.e. half of 1794 kg, is supported by the suspension
system on one side of the vehicle.
Develop a system of ordinary differential equations to describe the performance of the half vehicle model as the vehicle travels at a constant speed of 60 km/hr over a certain road surface. Choose your own model for the road surface but be sure to include a seam, i.e. a sharp step, some undulation and a stochastic (i.e. random) component. In this regard you may find the MATLAB
command and to be useful. You will find a considerable amount of information concerning road surfaces online. Animate the movement of the vehicle as it travels along the road.
Problem 3: Space-shuttle
It is a little difficult to obtain very precise information concerning space shuttle missions. Nonetheless we wish to model mission STS-30, the 29th shuttle mission, whose main objective was to deploy the Magellan planetary probe. Using the information available concerning the mission you are to mathematically model the craft (Atlantis) and produce an animation of the flight of the vehicle from launch to the point where the external tank is jettisoned, i.e. 8 minutes and 38 seconds into theflight. Although a roll manoeuvre is executed this is not required.
Of great significance in rocketry is the maximum-Q or max-Q. The dynamic pressure on the vehicle is equal to 1/2pv^2, where p is the air-density, which is decreasing as the vehicle climbs, and v is the velocity which is increasing as the vehicle accelerates. Evidently this quantity will achieve a maximum value at some point in the flight and this is referred to as max-Q.
Approaching max-Q is a time of particular concern for ground control and crew as it is the period where stress on the vehicle is at its highest and where the structure of the vehicle is most likely to fail. Accordingly at some time in advance of max-Q the engines are throttled down. For the space shuttle missions max-Q tended to occur at about 1 minute in not long after the craft had gone supersonic. In the case of mission STS-30 the anticipated max-Q time was 59 seconds into the flight.
We will assume the timing of events to be as scheduled. However, the inadequacies of our model may mean that the velocity and altitude are not in accord with the schedule above. You may model as you please. Highly simplified models will of course earn fewer grade steps. To earn all grade steps you may wish to bear in mind at least many, if not all, of the following issues:
At launch (a.k.a. SRB ignition) the orbiter and cargo are 217,513 lbs with the Magellan probe and
booster comprising most of the payload. Over the course of the simulated flight the shuttle climbs to a significant altitude of about 110 km. Its distance from the centre of the earth increases by about 1.73% of the earth’s radius. Accordingly
the acceleration due to gravity is not constant over the course of the flight, decreasing by a little over 3.37% The main engines can throttle between 67% and 111% of rated power in increments of 1% and the exact throttle value at all times is not known. We are told the throttle setting at three times, so the pragmatic assumption is to take the throttle value to be constant until a time where we are told that it is changed. Each of the three main engines uses about 350 US gallons of propellant per second at
100% throttle. You might assume a linear relationship between throttle and fuel efficiency for throttle between 67% and 111% , the operating range. At 100% throttle each engine produces 383,000 lb of force at sea level and 470,000 lb of force in vacuum. You will have to select some means of modelling the variation of thrust with altitude.
The propellant for the main engines was stored in the external tank. At launch 143,060 US gallons of liquid oxygen at a pressure of 22 psig were stored in the forward LOX tank and 383,066 US gallons of liquid hydrogen at a pressure of 29.3 psig were stored in the aft LH2 tank. Show that these figures yield 1,387,457 lb of liquid oxygen and 234,265 lb of liquid hydrogen. Assume the external tank is 78,100 lb when empty. I should note that these numbers are not consistent with total
weight of the vehicle as specified in the press kit for the mission, but they are the best numbers
which I can find.
The solid rocket boosters carry 1,107,000 lbs of solid rocket fuel each at lift off. Each booster is 193,000 lbs when empty. Assume the boosters burn fuel at a constant rate until it is completely consumed 1 second before Solid Rocket Booster staging. Assume also a constant thrust of 2,650,000 lb of force from each booster.
Problem 4: Mechanical linkage reprieve
Create a model for the linkage system of the backhoe of a JCB (or similar) digger. Analyse the system presenting equations to describe the performance. Hence simulate and visualise by producing a three-dimensional animation of the mechanism. Significant additional information concerning the problem may be found by referring to the earlier 10 bar linkage for the Ford
Mustang convertible. You may choose either to model the dynamical effects of motors, pumps, etc or to treat the problem as pseudo-static, in the manner of the previous problem. You should however not restrict the problem to a plane since this is unrealistic.
Engineering Assignment Help
TASK: Problem 1: A simple power supply is shown in Figure 1.
The input voltage signal is a single phase, 230 V (rms), 50 Hz sinusoid denoted “Mains Voltage” in Figure 2.1. Assume the four signal diodes in the bridge rectifier to be identical. Assume that they can be reasonably approximated by an ideal diode in series with a small bulk resistance, i.e. for a (forward bias) voltage v across one of these diodes the resulting (forward) current i flowing through . Is is the reverse saturation current, n is the ideality factor, V T is the thermal voltage (= kT/q where k is Planck’s constant, q is the magnitude of the electron charge and T is the temperature in K, i.e. in degrees Kelvin) and R Bulk is the bulk resistance. Appropriate parameter values for this model of the signal diode ar Assume a temperature of 300 K. By any means you choose determine a piecewise linear approximation to the DC characteristic of the signal diodes and employ this approximation in your simulation. In order to do this well you will need to make some credible estimate of the maximum current which could be flowing through a signal diode in forward bias. Your piecewise linear approximation need only be and should only be valid over a current range up to perhaps twice this value.
To keep things somewhat simple assume an ideal transformer with a turns ratio of 20:1. Again to keep things slightly simple model the Zener diode in the reverse bias region as an ideal Zener diode V Lwith a Zener breakdown voltage of 6.5 V in series with a very low resistance in the breakdown region of just 5.3 Ohm.
The electrolytic smoothing capacitor is rather large, C ? 2 , 200 ? F being the nominal capacitance. The tolerance of this capacitor is 20% The resistor R s is a 10% tolerance, 270 ? nominal, 1?2 W carbon film resistor. The load being supplied by the power supply is modelled by its Thévenin equivalent circuit. As the load generates no power this equivalent circuit consists of an input impedance but no voltage source. The load has several modes, depending on which devices are on or off and accordingly the impedance varies. There are two devices attached. One offers an inductive load and one a purely resistive load. Accordingly the load can be modelled as indicated in Figure
Derive equations describing the performance of the circuit for all four modes of the load (namely both devices off, both devices on and one device on. You may assume that the power supply is switched on at time t = 0, but that it has been off for a reasonable amount of time prior to this.
Accordingly, the voltage across the smoothing capacitor and the current through the inductor in the load have both had the time to decay to zero so that they may both be assumed to equal zero at time t = 0. For each of the four modes determine the resulting load voltage (denoted V L in Figure 2.1) for several cycles of the input voltage (sufficiently many that transients have passed). Given the precision of the smoothing capacitor and the resistance R S what range of steady-state load voltages do you expect to see? What is the maximum power dissipated in resistor R S ? Do you think that a 1?2
W power rating for this resistor is high enough?
For top credit (i.e. A or A+ grade) determine the load voltage for the following scenario: the power supply is switched on at time t = 0 at which time the smoothing capacitor voltage and inductor current are zero, neither attached loading device is active at this time; after 2 sec the resistive load is switched on; after a further 2.2 sec the inductive load is switched on and it remains active for the next 1.263 sec; subsequently the inductive device is switched off and remains off.
Problem 2: Vehicle Suspension
A vehicle is assumed to be completely symmetric about its central axis, i.e. the left hand side is the exact mirror image of the right hand side. A model for the suspension system of half of the vehicle is shown in Figure 3. The following parameter values have been deemed appropriate:
The wheelbase a+b and weight are approximately the values for a sixth generation BMW 5-series. Note that in the model half of the sprung mass, i.e. half of 1794 kg, is supported by the suspension
system on one side of the vehicle.
Develop a system of ordinary differential equations to describe the performance of the half vehicle model as the vehicle travels at a constant speed of 60 km/hr over a certain road surface. Choose your own model for the road surface but be sure to include a seam, i.e. a sharp step, some undulation and a stochastic (i.e. random) component. In this regard you may find the MATLAB
command and to be useful. You will find a considerable amount of information concerning road surfaces online. Animate the movement of the vehicle as it travels along the road.
Problem 3: Space-shuttle
It is a little difficult to obtain very precise information concerning space shuttle missions. Nonetheless we wish to model mission STS-30, the 29th shuttle mission, whose main objective was to deploy the Magellan planetary probe. Using the information available concerning the mission you are to mathematically model the craft (Atlantis) and produce an animation of the flight of the vehicle from launch to the point where the external tank is jettisoned, i.e. 8 minutes and 38 seconds into theflight. Although a roll manoeuvre is executed this is not required.
Of great significance in rocketry is the maximum-Q or max-Q. The dynamic pressure on the vehicle is equal to 1/2pv^2, where p is the air-density, which is decreasing as the vehicle climbs, and v is the velocity which is increasing as the vehicle accelerates. Evidently this quantity will achieve a maximum value at some point in the flight and this is referred to as max-Q.
Approaching max-Q is a time of particular concern for ground control and crew as it is the period where stress on the vehicle is at its highest and where the structure of the vehicle is most likely to fail. Accordingly at some time in advance of max-Q the engines are throttled down. For the space shuttle missions max-Q tended to occur at about 1 minute in not long after the craft had gone supersonic. In the case of mission STS-30 the anticipated max-Q time was 59 seconds into the flight.
We will assume the timing of events to be as scheduled. However, the inadequacies of our model may mean that the velocity and altitude are not in accord with the schedule above. You may model as you please. Highly simplified models will of course earn fewer grade steps. To earn all grade steps you may wish to bear in mind at least many, if not all, of the following issues:
At launch (a.k.a. SRB ignition) the orbiter and cargo are 217,513 lbs with the Magellan probe and
booster comprising most of the payload. Over the course of the simulated flight the shuttle climbs to a significant altitude of about 110 km. Its distance from the centre of the earth increases by about 1.73% of the earth’s radius. Accordingly
the acceleration due to gravity is not constant over the course of the flight, decreasing by a little over 3.37% The main engines can throttle between 67% and 111% of rated power in increments of 1% and the exact throttle value at all times is not known. We are told the throttle setting at three times, so the pragmatic assumption is to take the throttle value to be constant until a time where we are told that it is changed. Each of the three main engines uses about 350 US gallons of propellant per second at
100% throttle. You might assume a linear relationship between throttle and fuel efficiency for throttle between 67% and 111% , the operating range. At 100% throttle each engine produces 383,000 lb of force at sea level and 470,000 lb of force in vacuum. You will have to select some means of modelling the variation of thrust with altitude.
The propellant for the main engines was stored in the external tank. At launch 143,060 US gallons of liquid oxygen at a pressure of 22 psig were stored in the forward LOX tank and 383,066 US gallons of liquid hydrogen at a pressure of 29.3 psig were stored in the aft LH2 tank. Show that these figures yield 1,387,457 lb of liquid oxygen and 234,265 lb of liquid hydrogen. Assume the external tank is 78,100 lb when empty. I should note that these numbers are not consistent with total
weight of the vehicle as specified in the press kit for the mission, but they are the best numbers
which I can find.
The solid rocket boosters carry 1,107,000 lbs of solid rocket fuel each at lift off. Each booster is 193,000 lbs when empty. Assume the boosters burn fuel at a constant rate until it is completely consumed 1 second before Solid Rocket Booster staging. Assume also a constant thrust of 2,650,000 lb of force from each booster.
Problem 4: Mechanical linkage reprieve
Create a model for the linkage system of the backhoe of a JCB (or similar) digger. Analyse the system presenting equations to describe the performance. Hence simulate and visualise by producing a three-dimensional animation of the mechanism. Significant additional information concerning the problem may be found by referring to the earlier 10 bar linkage for the Ford
Mustang convertible. You may choose either to model the dynamical effects of motors, pumps, etc or to treat the problem as pseudo-static, in the manner of the previous problem. You should however not restrict the problem to a plane since this is unrealistic.
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