many students face difficulty while solving problems on translational mechanical systems in this short article lets find out how to easily calculate the required transfer function easily without any errors this method is already discussed in the book of control systems engineering by Norman s nise but lets simplify it with a answer template so that an average student can do it easily

step 1 : find out how many displacement variables are present in the given problem i.e 2 or 3 or more

step 2 : identify on which displacement or mass the force is acting on

step 3 : build the following table by looking at the given mechanical system system(for a 3 variable system)

displacement | sum of impedances connected to x1 or between | sum of impedances connected to x2 or between | sum of impedances connected to x3 or between |

x1 | (sum of all the impedances connected to x1) | -(sum of all the impedances connected between x1 and x2) | -(sum of all the impedances connected between x1 and x3) |

x2 | -(sum of all the impedances connected between x1 and x2) | (sum of all the impedances connected to x2) | -(sum of all the impedances connected between x2 and x3) |

x3 | -(sum of all the impedances connected between x1 and x3) | -(sum of all the impedances connected between x2 and x3) | (sum of all the impedances connected to x3) |

place the sum of impedances with positive sign if connected to x1 ,x2 or x3 and put it with negative sign if the impedances connected in between two displacements just like if connected in between x1 and x2 or x1 and x3.

step 4: calculate the determinant of this matrix written above and denote it as Δ

step 6 : write a single column matrix F clearly indicating on which mass the force is acting on

step 6 : now to calculate the transfer function Xi(s)/F(s) replace the i^{th }column of the above matrix with column matrix of F and calculate its determinant lets say this Δ_{i}

step 7 now the required transfer function is Δ_{i}/Δ

some of the solved problems you can find here https://electricalstudent.com/question-tag/translational-system-transfer-function/