Homework Help is Here – Start Your Trial Now!
Science
Advanced Physics
Vehicle Suspension System Spring constant k Passenger Fix Base with mass 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 as dt? dx + a3x = 0. + a2 dt Write down the values of a1, a2, 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!

Vehicle Suspension System Spring constant k Passenger Fix Base with mass 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 as dt? dx + a3x = 0. + a2 dt Write down the values of a1, a2, 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!

icon
Related questions
Question
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: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!
Vehicle Suspension System Spring constant k Passenger Fix Base with mass т Damping constant e 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 dx + azx = 0. + a2 dt a1 dt2 Write down the values of a1, a2, and a3 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?
Transcribed Image Text:Vehicle Suspension System Spring constant k Passenger Fix Base with mass т Damping constant e 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 dx + azx = 0. + a2 dt a1 dt2 Write down the values of a1, a2, and a3 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?
Expert Solution
steps

Step by step

Solved in 6 steps

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer