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Consider a safety suspension system designed to protect passengers from an impact in the event of a vehicle accident as shown in the above figure. The suspension system can be modelled as a spring-damper system with spring coefficient k and damping constant c. The passenger can be modelled as a point-mass with mass m and we assume that he/she uses a seatbelt in such a way that his/her body is always connected to the suspension system. We will use the concept of differential equation to predict the behaviour of this suspension system under various conditions.

Consider a safety suspension system designed to protect passengers from an impact in the event of a vehicle accident as shown in the above figure. The suspension system can be modelled as a spring-damper system with spring coefficient k and damping constant c. The passenger can be modelled as a point-mass with mass m and we assume that he/she uses a seatbelt in such a way that his/her body is always connected to the suspension system. We will use the concept of differential equation to predict the behaviour of this suspension system under various conditions.

Elements Of Electromagnetics
Elements Of Electromagnetics
7th Edition
ISBN:9780190698614
Author:Sadiku, Matthew N. O.
Publisher:Sadiku, Matthew N. O.
ChapterMA: Math Assessment
Section: Chapter Questions
Problem 1.1MA
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HELP!!! ASAP!!! DIFERENTIAL EQUATION

Case 2: We assume that the passenger is passive like in Case 1. However, this time, the damping mechanism is broken in such a way that the suspension only relies on the spring. g) Modify your answer in point a to account for the broken damper (c=0). h) Write down the characteristic equation of the differential equation in point g! i) Find the roots of the characteristic equation in point h! Use m=100 kg! j) Write down the position of the passengers x(t) if the passenger was initially at x(0) = 0 moving to the left with speed 10 m/s! k) Will the movement of the passenger stabilize over time (x(t) goes to zero)? If not, is it possible to make it stable by modifying the spring constant k? 1) How long would it take for the passenger to pass through his/her initial position 10 times? (Do not count the initial moment when x(0)=0)! Case 3: Now, with the damper still broken, the vehicle is experiencing a vibration in such a way that an external sinusoidal force F is experienced by the passenger where F = Acos(wt). Here, the amplitude A equals the last digit of your university student number multiplied by 100 (Nomor Induk Universitas/NIU) and the angular frequency w equals the last digit of your student number (Nomor Induk Universitas/NIU). m) Modify your answer in point g to account for the external force F and write down the non-homogeneous differential equation! n) Find the complementary solution x(1) to the differential equation in point m! Use the mass of the first passenger, m=100 kg. (Hint: complementary solution refers to the solution of the homogeneous version of differential equation in point m!) o) Find the particular solution x,(1) to the differential equation in point m! p) From your answer in point n and o, find the position of the passenger x(t) if the passenger was initially at x(0) = 0 moving to the left with speed 10 m/s! q) Now, if the angular frequency of the external force w is modified to w = prove that the particular solution x,(t) to the non-homogeneous differential equation will be Xp(t) = Aot sin (t). Find the value of Ao in terms of A, k, and m! Calculate the numerical value of Ao! r) Will the movement of the passenger stabilize over time (x(t) goes to zero)? If not, is it possible to make it stable by modifying the spring constant k? Explain your answer!
Transcribed Image Text:Case 2: We assume that the passenger is passive like in Case 1. However, this time, the damping mechanism is broken in such a way that the suspension only relies on the spring. g) Modify your answer in point a to account for the broken damper (c=0). h) Write down the characteristic equation of the differential equation in point g! i) Find the roots of the characteristic equation in point h! Use m=100 kg! j) Write down the position of the passengers x(t) if the passenger was initially at x(0) = 0 moving to the left with speed 10 m/s! k) Will the movement of the passenger stabilize over time (x(t) goes to zero)? If not, is it possible to make it stable by modifying the spring constant k? 1) How long would it take for the passenger to pass through his/her initial position 10 times? (Do not count the initial moment when x(0)=0)! Case 3: Now, with the damper still broken, the vehicle is experiencing a vibration in such a way that an external sinusoidal force F is experienced by the passenger where F = Acos(wt). Here, the amplitude A equals the last digit of your university student number multiplied by 100 (Nomor Induk Universitas/NIU) and the angular frequency w equals the last digit of your student number (Nomor Induk Universitas/NIU). m) Modify your answer in point g to account for the external force F and write down the non-homogeneous differential equation! n) Find the complementary solution x(1) to the differential equation in point m! Use the mass of the first passenger, m=100 kg. (Hint: complementary solution refers to the solution of the homogeneous version of differential equation in point m!) o) Find the particular solution x,(1) to the differential equation in point m! p) From your answer in point n and o, find the position of the passenger x(t) if the passenger was initially at x(0) = 0 moving to the left with speed 10 m/s! q) Now, if the angular frequency of the external force w is modified to w = prove that the particular solution x,(t) to the non-homogeneous differential equation will be Xp(t) = Aot sin (t). Find the value of Ao in terms of A, k, and m! Calculate the numerical value of Ao! r) Will the movement of the passenger stabilize over time (x(t) goes to zero)? If not, is it possible to make it stable by modifying the spring constant k? Explain your answer!
2. [LO2] Vehicle Suspension System Spring constant k Passenger with mass Fix Base т Damping constant c Consider a safety suspension system designed to protect passengers from an impact in the event of a vehicle accident as shown in the above figure. The suspension system can be modelled as a spring-damper system with spring coefficient k and damping constant c. The passenger can be modelled as a point-mass with mass m and we assume that he/she uses a seatbelt in such a way that his/her body is always connected to the suspension system. We will use the concept of differential equation to predict the behaviour of this suspension system under various conditions. Case 1: We assume that the passenger is passive and apply no external force to the system. a) Use Newton's Second Law to write down the passenger's equation of motion in the form of a second order differential equation as follows d²x + az + azx = 0. dx dt² Write down the values of a,, az, and a; in terms of m, k, and c! b) Write down the characteristic polynomial of the differential equation in point a! c) Suppose that c=400 kg/s, k=500 N/m. Find the roots of the characteristic equation for 3 different passengers with mass m=100 kg, m=60 kg, and m=80 kg! d) Write down the position of the previous 3 different passengers x(t) if the passenger was initially at x(0) = 0 m, moving to the left with speed 10 m/s! Will the movement of the passenger stabilize over time in the previous 3 cases? e) The first passenger (with mass m=100 kg) complains that his/her suspension system oscillates too much. He/she wants you to modify his/her suspension to behave similarly to the behaviour experienced by passenger 2 (with mass m=60 kg). Can you modify the spring constant k and the damping constant c to fulfil the request of passenger 1? f) Is it possible that for a certain passenger with mass M, the suspension system will fail to suppress the movement such that the passenger keeps getting faster? Explain your answer!
Transcribed Image Text:2. [LO2] Vehicle Suspension System Spring constant k Passenger with mass Fix Base т Damping constant c Consider a safety suspension system designed to protect passengers from an impact in the event of a vehicle accident as shown in the above figure. The suspension system can be modelled as a spring-damper system with spring coefficient k and damping constant c. The passenger can be modelled as a point-mass with mass m and we assume that he/she uses a seatbelt in such a way that his/her body is always connected to the suspension system. We will use the concept of differential equation to predict the behaviour of this suspension system under various conditions. Case 1: We assume that the passenger is passive and apply no external force to the system. a) Use Newton's Second Law to write down the passenger's equation of motion in the form of a second order differential equation as follows d²x + az + azx = 0. dx dt² Write down the values of a,, az, and a; in terms of m, k, and c! b) Write down the characteristic polynomial of the differential equation in point a! c) Suppose that c=400 kg/s, k=500 N/m. Find the roots of the characteristic equation for 3 different passengers with mass m=100 kg, m=60 kg, and m=80 kg! d) Write down the position of the previous 3 different passengers x(t) if the passenger was initially at x(0) = 0 m, moving to the left with speed 10 m/s! Will the movement of the passenger stabilize over time in the previous 3 cases? e) The first passenger (with mass m=100 kg) complains that his/her suspension system oscillates too much. He/she wants you to modify his/her suspension to behave similarly to the behaviour experienced by passenger 2 (with mass m=60 kg). Can you modify the spring constant k and the damping constant c to fulfil the request of passenger 1? f) Is it possible that for a certain passenger with mass M, the suspension system will fail to suppress the movement such that the passenger keeps getting faster? Explain your answer!
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