Oscillations of a vehicle driving on a sinewave ground

[J_R]

Homework Statement

In my homework I have to determine how do the speed (constant speed) (V) of damped mass and geometric characterictis of ground (b, h) influence on oscillation values of a model represented on a picture below. Model represents a vehicle driving on a sine wave ground. I should determine values of accelerations, velocities and displacements and explain why the results are as they are at different speeds and at different geometric characteristics.

k ... spring constant
d ... damping coefficient
h ... amplitude of a sine function (ground)
2*b ... period of a sine function (ground)
m ... mass ( of a vehicle)

y ... absolute displacement of a mass
z ... relative displacement of a mass

Homework Equations

$$y_0(x)=h*sin(\frac{pi*x}{b})$$
$$y_0(t)=h*sin(\frac{pi*V}{b}*t)$$
$$z(t)=y(t)-y_0(t)$$
$$m*\frac{d^2y}{dt^2}=-d*(\frac{dy}{dt}-\frac{dy_0}{dt})-k*(y-y_0)$$
$$m*\frac{d^2z}{dt^2}+d*\frac{dz}{dt}+k*z=-m*\frac{d^2y_0}{dt^2}$$

+ other equations that i don't think are relevant for what i am about to ask

The Attempt at a Solution

So, I have solved differential equations and got relative displacements, velocities and displacements of a mass. But for high speeds V and short periods of sine function of the ground b, i got enormous values for accelerations, and displacements equal the amlpitudes of sine function of the ground.

My guess is this happens because the mass cannot follow the ground at such high speeds and therefore lifts-off of the ground.

My questions are:

Is my thinking correct?
How can i determine the contact force, so that i would see when its value changes from + to - ; that is why the mass lifts-off, right?


Thank you for your help.

 

[rude man]

The mass should not lift off the ground. If it does, the problem becomes very difficult to solve. So assume there is always contact between m and the ground.

Assume the spring is initially relaxed at h = 0.

Your equations will also need to incorporate v, the lateral velocity.

 

[J_R]

I am aware that mass should not lift off ground. That is why i am trying to determine when (at what speed v and/or at what period b) that happens. Because when that happens i think that this model is not appropriate anymore. Is it possible to determine that with values of contact force? If so, how do i set an equation for that force?

V is incorporated in equation that defines y0, do i have to incorporate it somewhere else as well?

Rude man, thank you for your reply.

 

[rude man]

J_R, this needs more thinking on my part.
You have th right idea in writing your diff. eq. per my'' = ƩF. Obviously, two of the forces are the spriong and the damper. I need to think about the effect of the sinusoidal ground on thee equation.

Not a trivial problem, least for me. Maybe one of our powerhouse mech. types will help out.

 

[rude man]

I'm sorry, my later post got lost in the shuffle.

I was saying that you expression for z looks right except you left out gravity. Repeat the solution with gravity.

As to when the wheel leaves the ground: that point is when the ground force on the wheel = 0. So consider your forces on the wheel: gravity, spring, damper and the ground.

 

[J_R]

Thank you again. I will try that. When i took in account only spring and damper i got in each case wheel lifting of the ground at some point. With gravity included, i guess this will change.