A mass spring system is driven by the external force fext t...

A mass-spring system is driven by the external force Fext (t) = 2 sin (3t) + 2 cos (3t). The mass m=1, the spring constant k = 5 , and the damping coefficient b = 4. If the mass is initially located at y(0) = 0, with inital velocity y'(0) =2, find its equation of motion.

Transcript

00:02 Hi from the question it is given that a mass spring system is driven by the external force so it is given that the external force will be equal to 2 sine 3 t plus 2 cost 3 t and the mass is equal to 1 and the spring constant k is equal to 5 and the damping coefficient b is equal to 4 if the mass is initially located at y of 0 is equal to 0 with initial velocity y dash of 0 is equal to 2 so here we need to find the equation of motion.

00:34 So first consider we know that the formula for external force will be equal to m y double dash d y dash plus k y.

00:42 Here d is nothing but damping coefficient.

00:45 Instead of d we have considered as b.

00:48 So substitute the all the values given in the question.

00:51 So we have get 2 sine 3t plus 2 cost 3t will be equal to for m substitute 1 so we get y double dash plus for damping coefficient is 4 so 4 y dash plus and the spring constant is 5, so 5y.

01:09 So let us assume that y is equal to e power mt, so y dash will be equal to m e par mt and y double dash is equal to m d '2.

01:17 Now substitute all these values in the above equation, so we get m squared e power m plus 4 eme power mt plus 5 e power mt is equal to 0.

01:28 Now remove the common term so then the above expression became m square plus 4m plus 5 is equal to 0.

01:34 Now solve this by using the quadratic equation we know that the quadratic equation equation is nothing but minus b plus or minus square root of b squared minus 4 a c divided by 2a so apply this formula so we have get minus 4 plus or minus square root of 16 minus 4 into a is 1 and c is 5 divided by 2a a is 1 therefore is just a 2 and again simplify this so we get minus 4 plus or minus 16 minus 20 is minus 4 divided by 2 and the value of square root of minus 1 is i therefore minus 4 plus or minus i into 2 divided by 2 which is will be equal to minus 2 plus or minus i and the characteristic equation can be represented as y of t is equal to e to the power of minus 2 t into a sine t plus b coste and let this be equation number 1 this can be arrived by using this formula now the particular solution will be equal to yp which will be equal to c into cost 3t plus d sine 3t.

02:46 Now differentiate with respect to t...

Question

A mass-spring system is driven by the external force Fext (t) = 2 sin (3t) + 2 cos (3t). The mass m=1, the spring constant k = 5 , and the damping coefficient b = 4. If the mass is initially located at y(0) = 0, with inital velocity y'(0) =2, find its equation of motion.

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