Mechanical vibration-car suspension system over series of hills

In summary: I solved the system using the equation for a sinusoidal spring/mass/damper system with an applied external force.
  • #1
kingclayton8
2
0
I have a final exam in a mechanical vibrations course this week and I know what one of the problems might be like. It consists of a spring/mass/damper system (like a car supsension) that travels over a series of hills in a sinusoidal pattern. The car suspension has a unique coordinate system (Y) and the wheel at the base of the system has a unique coordinate system (X) and both move differently. The problem would involve analyzing it somehow. I wish I could draw the picture. I can't seem to find anything like it in the textbook or the internet. My professor is crazy about Conservation of Energy when analyzing systems like this. Any Ideas? Any help would be greatly appreciated!
 
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  • #2
Most of the following discussion should be familiar to you. I am writing it here for completeness. Also, it’s a good review for me.

Assume one spring/mass/damper suspension system is set up vertically under a car with one wheel. Assume also that the system is bolted to the car and the wheel stays on the road. The y position of the wheel at any point in time is the elevation of the road (which changes over time) plus the radius of the wheel (which is constant). The y position of the car is the y position of the wheel plus the equilibrium height of the suspension (constant) plus the deflection of the suspension (changes over time).

You already know that when you first place the car on the road, the spring/mass/damper combined with the weight of the car will cause the elevation of the car to oscillate in smaller and smaller increments until it finally comes to rest at an equilibrium height above the road. You should have the derivation for this “damped oscillation under a constant external force” in your textbook.

If the road is flat, the y position of the wheel is constant. The y position of the car is solved when you fill in the values for all the variables the above mentioned equation. But since the road has hills, there is an additional force on the spring/mass/damper and that force is not constant. As the car goes up and down the hills the external force is equal to the weight of the car plus the force necessary to elevate/lower the car to keep it on the road. You can model this by thinking of the weight of the car as varying. When the car goes up a hill, it gets heavier. When it goes down a hill, it gets lighter.

So now you need to ask “what is the mathematical function for the varying force/weight on/of the car?”. The teacher is making it easy for you by specifying that the hills are sinusoidal in shape. Sinusoidal shaped hills means the vertical deflection of the wheel varies by the sine, the vertical velocity varies by the cosine and the vertical acceleration varies by the sine. Force is mass times acceleration. So the external force on the wheel as a result of the hills is the mass of the car times the sine function. The total external force on the wheels (apparent weight) is the real weight of the car plus this sine function. You can model this total external force as a single sine function. When you do that you have a spring/mass/damper system under a sinusoidal forcing function. You should have a generic derivation for that arrangement in your textbook too.

That’s the setup. If you need help with the actual equations I can go through them with you too. But I should really wait till after the exam for that.
 
  • #3
Thank you very much Mike. The final exam went well.
 

1. What is a mechanical vibration-car suspension system?

A mechanical vibration-car suspension system is a system that helps absorb the shock and impact of bumps and uneven surfaces while driving, providing a smoother and more comfortable ride for passengers.

2. How does a car suspension system work?

A car suspension system works by using a combination of springs, shocks, and other components to absorb the energy from the wheels when they hit bumps or uneven surfaces, preventing it from being transferred to the rest of the car and its occupants.

3. How does a car suspension system handle hills?

A car suspension system is designed to handle hills by adjusting the height and stiffness of the suspension components to maintain a stable and smooth ride. The system will also work to keep the wheels in contact with the road surface, preventing loss of traction and maintaining control.

4. What are the benefits of a well-designed car suspension system?

A well-designed car suspension system provides a number of benefits, including improved handling and control, increased comfort for passengers, and reduced wear and tear on the vehicle's tires and other components. It can also help improve fuel efficiency by reducing the amount of energy needed to maintain a steady ride.

5. How can a car suspension system be maintained?

A car suspension system can be maintained by regularly checking and replacing worn out or damaged components, such as shocks and springs. It is also important to have the system inspected and serviced by a professional mechanic on a regular basis to ensure it is functioning properly and to address any potential issues before they become more serious problems.

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