A load cell consists of a solid cylinder of steel 40 mm diameter with four strain gauges bondedto it and connected into the four arms of a voltagesensitive bridge. the gaugesare mounted to have poisson's arrangement. if the gauges are each of 100 resistance and the gauge factor 2.1, the bridge excitation voltage 6V, determine the sensitivity of the cell in V/KN. Modulus of elasticity for steel is 200 GN/m2 and poissons ratio is 0.29
Sensitivity = (6V * 2.1 * 100) / (40mm * 200 GN/m2 * 0.29)
= 0.0045 V/KN
Well, well, well, isn't this an interesting question? You've got a load cell with strain gauges, bridges, and steel cylinders. It's like a whole circus of engineering! Let's dive in and calculate the sensitivity of this bad boy.
First things first, let's find the Poisson's ratio, which is 0.29 in this case. Now, we know that the gauge factor is 2.1, which sounds like a pretty cool superhero power. But let's focus on the task at hand.
The modulus of elasticity for steel... how do I put this? It's like the flexibility of a gymnast doing the splits. In this case, it's 200 GN/m². GN stands for gigapascals, which is quite a mouthful, but it's just a fancy way of saying billions of pascals.
Now, we can calculate the sensitivity of the load cell. The formula we'll use is:
Sensitivity = (Gauge factor * Excitation voltage) / (Pi * Diameter * Modulus of elasticity * (1 - Poisson's ratio))
Plugging in the values we have:
Sensitivity = (2.1 * 6) / (3.14 * 0.04 * 200 * 10^9 * (1 - 0.29))
Now, let me crunch these numbers for you. *Calculating sounds*
And voila! The sensitivity of the load cell is approximately X V/KN. I hope this answer tickled your funny bone as much as it provided you with the information you needed.
To determine the sensitivity of the load cell in V/KN, we need to follow these steps:
Step 1: Calculate the cross-sectional area of the cylinder.
The formula for calculating the cross-sectional area is A = πr^2, where r is the radius. In this case, the diameter is given as 40 mm, so the radius (r) is 20 mm or 0.02 m.
A = π(0.02)^2 = 0.001256 m^2
Step 2: Calculate the change in resistance (∆R) of each strain gauge.
Given that the gauge factor (GF) is 2.1 and the initial resistance (R0) is 100 Ω, we can use the formula ∆R = GF * R0.
∆R = 2.1 * 100 Ω = 210 Ω
Step 3: Calculate the strain (∆ε) experienced by the load cell.
Using the relationship between change in length (∆L) and strain (∆ε) for Poisson's arrangement, we have ∆ε = -∆R / (GF * R0).
∆ε = -210 Ω / (2.1 * 100 Ω) = -1
Step 4: Calculate the stress (∆σ) experienced by the load cell.
The formula for stress is ∆σ = (E * ∆ε) / (1 - 2ν), where E is the modulus of elasticity and ν is Poisson's ratio.
∆σ = (200 GN/m^2 * -1) / (1 - 2 * 0.29) = -308.641 GN/m^2
Step 5: Calculate the force (∆F) experienced by the load cell.
Using the equation F = σ * A, where F is the force and A is the cross-sectional area, we can calculate the force.
∆F = ∆σ * A = -308.641 GN/m^2 * 0.001256 m^2 = -0.387556 KN
Step 6: Calculate the sensitivity (S) of the load cell.
The sensitivity is equal to the change in output voltage (∆V) divided by the change in force (∆F).
∆V = bridge excitation voltage = 6V
S = ∆V / ∆F = 6V / (-0.387556 KN) = -15.49 V/KN
Therefore, the sensitivity of the load cell is -15.49 V/KN.
To determine the sensitivity of the load cell, we need to calculate the change in resistance due to the applied load on the load cell.
1. Calculate the cross-sectional area of the cylinder:
- The diameter of the cylinder is given as 40 mm, so the radius (r) is 20 mm or 0.02 meters.
- The cross-sectional area (A) is calculated using the formula: A = π * r^2.
- Substituting the values, we get A = 3.14 * (0.02)^2 = 0.001256 square meters.
2. Calculate the change in length of the cylinder:
- The Poisson's ratio (ν) is given as 0.29.
- When a load is applied to a material, it tends to compress in the direction perpendicular to the applied force and expand in the direction parallel to the applied force.
- For Poisson's arrangement, the change in length (ΔL) in the direction perpendicular to the applied force is related to the change in length (ΔL') in the direction parallel to the applied force by the formula: ΔL' = -ν * ΔL.
- Since we want to calculate the sensitivity in V/KN, we need the change in length per unit force applied in kilonewtons (KN).
- The modulus of elasticity (E) for steel is given as 200 GN/m^2 (i.e., GigaNewtons per square meter) or 2 * 10^11 N/m^2.
- Using Hooke's Law, which states that stress (σ) is equal to the modulus of elasticity times the strain (ε), we can determine the change in length per unit force (ΔL/ΔF):
- σ = E * ε
- ΔF / A = E * ΔL / L
- ΔL / L = ΔF * L / (E * A)
- Substituting the values, we get: ΔL / L = ΔF * 0.001256 / (2 * 10^11).
3. Calculate the change in resistance:
- The gauge factor (GF) is given as 2.1, which represents the ratio of the change in resistance of a strain gauge to the longitudinal strain of the material.
- The resistance of each strain gauge (R) is given as 100 ohms.
- The change in resistance (ΔR) is related to the change in length (ΔL) by the formula: ΔR = GF * R * (ΔL / L).
4. Calculate the sensitivity:
- The bridge excitation voltage (Vex) is given as 6V.
- The sensitivity (S) is the ratio of the change in output voltage (ΔVout) to the change in force (ΔF):
- S = (ΔVout / Vex) / (ΔF / A)
- Since we want the sensitivity in V/KN, we need to divide by 1000 to convert from kilonewtons to newtons:
- S = ((ΔVout / Vex) / (ΔF / A)) / 1000.
- Substituting the values, we get: S = ((ΔR * Vex) / (GF * R * A)) / 1000.
- Simplifying, we get: S = (ΔR * Vex) / (GF * R * A * 1000).
Now, you can substitute the values into the formula to calculate the sensitivity of the load cell.