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Calculating the center of mass - with examples
Calculating the center of mass is an important step in many tasks in mechanical engineering and in the design of machines and components. The center of mass indicates where the weight of a body is concentrated and thus permits determining the forces and moments in the system. This article explores the basics of calculating the center of mass and provides some real-world examples.
What is the center of mass?
The center of mass or center of gravity is where the entire weight of a body is concentrated. It is determined by the location of all individual masses within the system and their distances to the point of origin.
The center of mass is the “point of attack ” for gravity. The object behaves like a point mass in the gravitational field.
Important - the center of mass may also be outside the body. For example, on hemispheric shells. A torque is ineffective when exerted at the center of gravity.
For homogeneous bodies (i.e., equal density everywhere), the center of mass corresponds to the geometric center of gravity (center of volume) - these bodies are so-called trivial individual masses. The center of gravity of homogeneous bodies is therefore easiest to determine.
The opposite of homogeneous bodies are so-called inhomogeneous bodies - they have different densities in the body sections. They cannot be considered single masses. Such bodies must be divided into suitable individual masses, calculated individually and ultimately reconciled into the entire system.
The center of mass calculation is important in many engineering applications.
An example is the design of a machine and its components: here, the center of gravity of the components must be selected such that the overall machine is stable and safe and whose components are properly “balanced”.
Methods for calculating the center of mass
There are various methods for determining the center of mass depending on the geometry and how the mass (densities) is distributed in the system.
- On homogeneous bodies, the center of volume can be selected as the center of gravity, provided all densities are distributed uniformly.
- For inhomogeneous bodies, the center of mass must be determined taking into account all point densities.
Generally, the center of gravity can be calculated as the sum of all sub-masses, multiplied by their respective distances to the origin, divided by the total mass. The body is broken down into a finite number of sub-quantities.
Modern CAD programs or FEM (finite element method) programs offer such calculation methods for the center of mass as standard features.
Center of mass and center of volume
The center of volume does not take into account mass or densities of the body. The center of volume is therefore a special case of the center of mass, given uniformly distributed density in the object.
The calculation of the center of mass can be simplified for homogeneous bodies.
Effort and utility of the calculations
A suitable division in individual masses is not always trivial - especially for non-uniformly distributed densities. Such problems can be solved computationally and experimentally. The accuracy of the result is expected to depend on the feasible calculation depth or the measurement accuracy. Results can only be approximated - effort and benefit should be therefore be weighed.
Center of mass for homogeneous bodies
For homogeneous bodies such as a cuboid or cylinder, the center of gravity can be easily determined by geometric considerations.
Symmetries can be used in this case to simplify the problem.
The center of mass matches the geometric center of gravity and is easily calculated. In this example, the center of mass is simultaneously the center of the circular area and the projected area of the rectangle.
Center of mass for irregularly shaped objects or inhomogeneous objects
For irregularly shaped objects, one must consider each point (point density) individually, and its contribution to the total mass must be calculated.
This approach is also called integration.
Polyhedron with evenly distributed density
The geometric center of gravity of the body is calculated by splitting the body into suitable partial bodies. The centers of gravity of these partial bodies are calculated and then weighted over the proportion of the area or volume.
The geometric center of gravity is the center of mass.
Polyhedron with unevenly distributed density
The geometric center of gravity of the body with unevenly distributed density is identical to the geometric center of gravity of the body with evenly distributed density.
The geometric center of gravity does not lie at the center of mass.
The body must be broken down into suitable partial bodies and their individual centers of gravity must be determined based on the shape and the unequally distributed density.
The center of mass is calculated from the partial bodies taking into account the body volume and body masses
(x_s,y_s,z_s) = \frac{1}{M}\sum_i(x_{si}, y_{si}, z_{si})\cdot m_i
- M - Total mass
- mi - Partial mass
- (xsi, ysi, zsi) - center of gravity coordinates of partial body 1 in the spatially fixed coordinate system (x, y, z)
- (xs, ys, zs) - center of gravity coordinates of the entire object in the spatially fixed coordinate system (x, y, z)
Explicit formula for center of mass
If one performs progressively finer brake downs, partial volumes or partial masses "approach zero". As a result, the above approximation formula is converted to an integral.
The center of gravity can thus be determined very precisely:
x_s=\frac{1}{M}\int \int_V \int \rho (x,y,z)xdV
y_s=\frac{1}{M}\int \int_V \int \rho (x,y,z)ydV
z_s=\frac{1}{M}\int \int_V \int \rho (x,y,z)zdV
- M - Total mass
- p(x, y, z) - Local density of the material
- V - Volume of the component
Center of mass for compound systems
Compound systems consist of several interconnected individual bodies that each have their own center of gravity.
To find the common center of gravity of all sub-objects, each of these points must be weighted with its corresponding mass.
Example calculation: Combined center of gravity of 2 subsystems
A system consisting of two distinct subsystems is combined into a combined center of gravity.
x_s=\frac{x_{s1}\times m_1+x_{s2}\times m_2}{m_1+m_2}
y_s=\frac{y_{s1}\times m_1+y_{s2}\times m_2}{m_1+m_2}
z_s=\frac{z_{s1}\times m_1+z_{s2}\times m_2}{m_1+m_2}
- m1 - Mass of partial body 1
- (xs1, ys1, zs1) - center of gravity coordinates of partial body 1 in the spatially fixed coordinate system (x, y, z)
- m2 - Mass of partial body 2
- (xs2, ys2, zs2) - center of gravity coordinates of the partial body 1 in the spatially fixed coordinate system (x, y, z)
- (xs, ys, zs) - center of gravity coordinates of the entire object in the spatially fixed coordinate system (x, y, z)
Determining center of mass experimentally
The center of mass can also be determined experimentally. Experimental measurement methods have some advantages over purely theoretical calculations:
- They are independent of the material model,
- they automatically consider all sources of error,
- they provide a direct measurement that is not dependent on assumptions or estimates.
Oscillation Method
The oscillation method is based on the principle of harmonic oscillation. This involves suspending an object on a thin wire and causing it to oscillate. The angular velocity can be calculated by measuring the period duration. The angular velocity can then be used to determine the distance between the suspension point and the center of mass.
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Scale method
This method places the object to be examined on a platform scale and measures its weight. The same procedure is then performed with a second weight to measure the distance between both points. Multiplying the weight force by the distance results in a moment equation for determining the center of mass.
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Tilt Method
The tilt method is based on the principle of static stability. The object to be examined is placed on a flat surface and tested for tilting by moving weights to different positions. The center of mass can then also be determined by determining the gravitational center line.
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