ENG1005 quarber car [Corrugated road in Western Australia. Author: Bras / CC BY-SA (http://creativecommons.org/licenses/by-sa/3.0/)] Simple descriptions...

 

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ENG1005 "quarber car [Corrugated road in Western Australia. Author: Bäras / CC BY-SA (http://creativecommons.org/licenses/by-sa/3.0/)] Simple descriptions of a car's suspension consider a single wheel supporting a quarter of the mass of the car. The suspension itself is modelled as a damped spring. This setup is sketched below. M Spring dashpot wheel ASSIGNMENT 4 Zroad = 0 x = 0 Zroad (x) = H|1 12H - COS = H[1-cos (²2²)] (2лх L Zroad=1 We will assume that the car is travelling at a horizontal speed V over the road. This speed is constant, but the driver can choose its value. For x < 0, the road is flat Zroad = 0 and for x > 0 there are corrugations given by 4 x where H is the height of each corrugation and L is its length. We will assume that time t = 0 occurs when the car just reaches x = 0. Then the equation governing the vertical displacement z(t) of the car body is mz" =-k[z-Zroad (Vt)] - Dz' where primes indicate differentiation with respect to time, m is the quarter mass of the car, D is the damping factor of the suspension and k is the spring constant. We will take m = 1/2, D = 5, k = 37, H = 0.1, L = 0.2 (these are crudely reasonable numbers if masses are measured in tonnes, lengths in metres and time in seconds). We will explore the response for different speeds V. (5) Plot graphs of the particular solutions of this ODE for a variety of values of a. Be sure to include relatively slow and relatively fast speeds. [3 marks] (6) Describe the amplitudes of the solutions plotted in the last question. How does the speed affect the amplitude? [2 marks] α = 2лV L Solve for the particular solution to this differential equation. You should show that the particular solution is given by b= z(t) = bcos(at) + csin(at) + H where the constants b and c are given by -(k-a² m) kH and (k-a²m)² + (aD)² c= -aD kH (k-a² m)² + (aD)²* ENG1005 "quarber car [Corrugated road in Western Australia. Author: Bäras / CC BY-SA (http://creativecommons.org/licenses/by-sa/3.0/)] Simple descriptions of a car's suspension consider a single wheel supporting a quarter of the mass of the car. The suspension itself is modelled as a damped spring. This setup is sketched below. M Spring dashpot wheel ASSIGNMENT 4 Zroad = 0 x = 0 Zroad (x) = H|1 12H - COS = H[1-cos (²2²)] (2лх L Zroad=1 We will assume that the car is travelling at a horizontal speed V over the road. This speed is constant, but the driver can choose its value. For x < 0, the road is flat Zroad = 0 and for x > 0 there are corrugations given by 4 x where H is the height of each corrugation and L is its length. We will assume that time t = 0 occurs when the car just reaches x = 0. Then the equation governing the vertical displacement z(t) of the car body is mz" =-k[z-Zroad (Vt)] - Dz' where primes indicate differentiation with respect to time, m is the quarter mass of the car, D is the damping factor of the suspension and k is the spring constant. We will take m = 1/2, D = 5, k = 37, H = 0.1, L = 0.2 (these are crudely reasonable numbers if masses are measured in tonnes, lengths in metres and time in seconds). We will explore the response for different speeds V. (5) Plot graphs of the particular solutions of this ODE for a variety of values of a. Be sure to include relatively slow and relatively fast speeds. [3 marks] (6) Describe the amplitudes of the solutions plotted in the last question. How does the speed affect the amplitude? [2 marks] α = 2лV L Solve for the particular solution to this differential equation. You should show that the particular solution is given by b= z(t) = bcos(at) + csin(at) + H where the constants b and c are given by -(k-a² m) kH and (k-a²m)² + (aD)² c= -aD kH (k-a² m)² + (aD)²*

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1 1 1H Proad K27 without loss of generality force in the spring m we Osume ZZ rood k vextension k ... View the full answer