# How to calculate velocity in a spring mass model?

1. Aug 9, 2012

### Noblesse

1. The problem statement, all variables and given/known data

At time t = 0, the mass is released 8 mm below the static equilibrium position. The mass is
m = 4kg and each spring stiffness is k = 40 kN/m.
Determine:
(i) The equivalent spring stiffness.
(ii) The natural frequency of the system in Hertz.
(iii) The displacement equation of the spring-mass model.
(iv) The velocity of the mass at time t = 0.05s.
(v) The acceleration of the mass at time t = 0.05s.

2. Relevant equations

x=A sin⁡〖ω_n 〗 t+B cos⁡〖ω_n 〗 t

3. The attempt at a solution I have managed to calculate it. Basically B is equal to displacement at t=0 according to lectures about undamped free vibration and A is equal to velocity x/natural circular frequency. In order to find velocity, I calculated natural circular frequency using the W=square root of (stiffness(k)/mass(m)) and after it I differentiated the formula for x given above in the relevant equations. Pretty much I just have to replace B for displacement and then multiplied by natural frequency in the derived velocity formula. A would be ignored and the value for B times W_n would be used as the value for velocity I believe. The derivation of the above formula should be something like this:

x=ω_n A cos⁡〖ω_n 〗t - ω_n B sin⁡〖ω_n 〗t

This is about undamped free vibration for Dynamics. If someone knows about it, would be nice if they could give me a shout and let me know if Im wrong and add some input into it here in the thread

Last edited: Aug 9, 2012
2. Aug 9, 2012

### Noblesse

Velocity found. For acceleration I need to differentiate the formula for velocity and use the same method I think.