> A spring-mass system is a fundamental concept in physics and engineering that describes the dynamic...

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> A spring-mass system is a fundamental concept in physics and engineering that describes the dynamic behavior of a mass attached to a spring. This system is commonly used to model and analyze various mechanical and oscillatory phenomena in engineering applications. Understanding spring-mass systems is crucial in fields such as automotive suspension systems, seismic isolation devices, vibration isolation in machinery, tuned mass dampers, energy harvesting devices etc. www Hooke's Law states that the force exerted by a spring is directly proportional to the displacement of the spring from its equilibrium or rest position. This relationship is expressed mathematically as given below. F = -kx The motion of the mass-spring system is governed by Newton's second law, which relates the force acting on an object to its mass and acceleration. By combining Newton's second law with Hooke's Law, a second-order linear differential equation is derived, providing a mathematical representation of the system's behavior. -Σkx + mg For static case in which acceleration term vanishes (a=0), remaining equation exhibits a system of linear equations for multiple spring-mass attached to each other as illustrated in Figure 5. m d²x dt² [k₁ + k₂ -k₂ 0 0 0 =ma =- 0 = m₁g + k₂(x₂-x₁) - K₁x₁ m₂g + k₂(x₁ -X₂) + K3(x3 - X₂) 0 = m3g + k3 (X2 X3) + K4(X4 - X3) 0 0= m4g + k4(X3 X4) + K5 (X5 - X4) 0= msg + k5(x4- Xx5) + k6(x6 - X5) 0= meg + ko(xs - x6) -K₂ k₂ + k3 -K3 0 0 0 0 -K3 k3 + K₁ -K₁ 0 0 0 0 -KA k₁ + ks 0 0 0 0 0 0 -ks 0 -ks ks + k6 -ko 0 -k6 k6 X₁5 X₁ X₂ X3 x₂√5 X5 X3√ X5√√ E-m6 x65 Figure 5. Spring-mass system 2. Example Application of Newtons 2nd Law and Hooke's Law on Spring-Mass System. The static solution of spring mass system illustrated in Figure5 can be considered by applying Newtons 2nd and Hooke's laws to each mass components separately. Linear equations for each mass is given below. wwwwww Therefore, the problem amounts to solving six equations with six unknown positions. These System of linear equations can be expressed in matrix form as. -m₁ mig m₂g m3g m4g m5g m6g/ m₂ -m3 -- m4 k₁ k₂ -ms k3 ks Ká 3. Assignment Consider spring-mass system shown in Figure 5 and solve for positions (x₁) in the system by MATLAB code with an iterative solution method for system of linear equations. Mass and spring stiffness values (mi to me and ki to k6) are different for each student as given in Table 2. b) Explain selected algorithm with a proper and readable flow chart. (5 Points) c) Encode MATLAB script for selected algorithm. (25 Points) d) Find positions as denoted in Figure 5 and comment on the results. (10 Points) e) Find three other engineering problems that can be illustrated with system of linear a) Select proper numerical method to solve the problem. Explain your reasons of method selection. (5 Points) g=-9.817 > A spring-mass system is a fundamental concept in physics and engineering that describes the dynamic behavior of a mass attached to a spring. This system is commonly used to model and analyze various mechanical and oscillatory phenomena in engineering applications. Understanding spring-mass systems is crucial in fields such as automotive suspension systems, seismic isolation devices, vibration isolation in machinery, tuned mass dampers, energy harvesting devices etc. www Hooke's Law states that the force exerted by a spring is directly proportional to the displacement of the spring from its equilibrium or rest position. This relationship is expressed mathematically as given below. F = -kx The motion of the mass-spring system is governed by Newton's second law, which relates the force acting on an object to its mass and acceleration. By combining Newton's second law with Hooke's Law, a second-order linear differential equation is derived, providing a mathematical representation of the system's behavior. -Σkx + mg For static case in which acceleration term vanishes (a=0), remaining equation exhibits a system of linear equations for multiple spring-mass attached to each other as illustrated in Figure 5. m d²x dt² [k₁ + k₂ -k₂ 0 0 0 =ma =- 0 = m₁g + k₂(x₂-x₁) - K₁x₁ m₂g + k₂(x₁ -X₂) + K3(x3 - X₂) 0 = m3g + k3 (X2 X3) + K4(X4 - X3) 0 0= m4g + k4(X3 X4) + K5 (X5 - X4) 0= msg + k5(x4- Xx5) + k6(x6 - X5) 0= meg + ko(xs - x6) -K₂ k₂ + k3 -K3 0 0 0 0 -K3 k3 + K₁ -K₁ 0 0 0 0 -KA k₁ + ks 0 0 0 0 0 0 -ks 0 -ks ks + k6 -ko 0 -k6 k6 X₁5 X₁ X₂ X3 x₂√5 X5 X3√ X5√√ E-m6 x65 Figure 5. Spring-mass system 2. Example Application of Newtons 2nd Law and Hooke's Law on Spring-Mass System. The static solution of spring mass system illustrated in Figure5 can be considered by applying Newtons 2nd and Hooke's laws to each mass components separately. Linear equations for each mass is given below. wwwwww Therefore, the problem amounts to solving six equations with six unknown positions. These System of linear equations can be expressed in matrix form as. -m₁ mig m₂g m3g m4g m5g m6g/ m₂ -m3 -- m4 k₁ k₂ -ms k3 ks Ká 3. Assignment Consider spring-mass system shown in Figure 5 and solve for positions (x₁) in the system by MATLAB code with an iterative solution method for system of linear equations. Mass and spring stiffness values (mi to me and ki to k6) are different for each student as given in Table 2. b) Explain selected algorithm with a proper and readable flow chart. (5 Points) c) Encode MATLAB script for selected algorithm. (25 Points) d) Find positions as denoted in Figure 5 and comment on the results. (10 Points) e) Find three other engineering problems that can be illustrated with system of linear a) Select proper numerical method to solve the problem. Explain your reasons of method selection. (5 Points) g=-9.817