Question
Consider two helical springs with the following characteristics:Spring 1: material $-$ steel; number of turns $-10 ;$ mean coil diameter $-0.3 \mathrm{~m} ;$ wire diameter$0.05 \mathrm{~m} ;$ free length $-0.4 \mathrm{~m} ;$ shear modulus $-80 \times 10^{9} \mathrm{~Pa}$.Spring 2: material-aluminum; number of turns $-10 ;$ mean coil diameter $-0.25 \mathrm{~m} ;$ wire diameter $-0.025 \mathrm{~m} ;$ free length $-0.4 \mathrm{~m} ;$ shear modulus $-24 \times 10^{9} \mathrm{~Pa}$. Determine the equivalent spring constant when (a) spring 2 is placed inside spring $1,$ and $(b)$ spring 2 is placed on top of spring 1 .
Step 1
Step 1: First, we calculate the spring constant for each spring using the formula: \[k = \frac{Gd^4}{64nR^3}\] where \(G\) is the shear modulus, \(d\) is the wire diameter, \(n\) is the number of turns, and \(R\) is the mean coil radius. Show more…
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Design a steel helical compression spring to satisfy the following requirements: Spring stiffness $(k) \geq 8000 \mathrm{~N} / \mathrm{mm}$ Fundamental natural frequency of vibration $\left(f_{1}\right) \geq 0.4 \mathrm{~Hz}$ Spring index $(D / d) \geq 6$ Number of active turns $(N) \geq 10$ The stiffness and fundamental natural frequency of the spring are given by [1.43]: $$ k=\frac{G d^{4}}{8 D^{3} N} \text { and } f_{1}=\frac{1}{2} \sqrt{\frac{k g}{W}} $$ where $G=$ shear modulus, $d=$ wire diameter, $D=$ coil diameter, $W=$ weight of the spring, and $g=$ acceleration due to gravity.
The Spring Constant for a helical coil spring, theoretically follows: k = Gd^4 / (8D^3na) where d is the wire diameter, D is the mean diameter of the coil, na is the number of active coils and G is a material property called the shear modulus or modulus of rigidity (see Fig. 2). The shear modulus, G, depends on the spring material and is estimated using text or online sources such as MatWeb (www.matweb.com).
The spring constant of a helical spring under axial load is given by $$ k=\frac{G d^{4}}{8 N D^{3}} $$ where $G$ is the shear modulus, $d$ is the wire diameter, $D$ is the coil diameter, and $N$ is the number of turns. Find the spring constant and the weight of a helical spring made of steel for the following data: $D=0.2 \mathrm{~m}, d=0.005 \mathrm{~m}, N=10$.
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